The base, we get the segment CE, the trapezoid is divided into two - the rectangle ABCE and the right triangle ECD. The hypotenuse is the lateral side we know trapeze CD, one of the legs is equal to the perpendicular side trapeze(according to the rectangle rule, two parallel sides are equal - AB = CE), and the other is a segment whose length of bases trapeze ED=AD-BC.

Find the legs of the triangle: using the existing formulas CE = CD*sin(ADC) and ED = CD*cos(ADC). Now calculate the upper base - BC = AD - ED = a - CD*cos(ADC) = a - d*cos (Alpha). Find out the length of the perpendicular side - AB \u003d CE \u003d d * sin (Alpha). So, you got the lengths of all sides of the rectangular trapeze.

Add the resulting values, this will be the perimeter of a rectangular trapeze:P = AB + BC + CD + AD = d*sin(Alpha) + (a - d*cos(Alpha)) + d + a = 2*a + d*(sin(Alpha) - cos(Alpha) + one).

Task 3. Find the perimeter of a rectangular trapeze, if the lengths of its bases are known AD = a, BC = c, the length of the perpendicular lateral side AB = b and sharp corner with the other side ADC = Alpha. Solution. Draw a perpendicular CE, get a rectangle ABCE and a triangle CED. Now find the length of the hypotenuse of the triangle CD = AB / sin (ADC) = b / sin (Alpha). So, you got the lengths of all sides.

Add up the resulting values: P = AB + BC + CD + AD = b + c + b/sin(Alpha) + a = a + b*(1+1/sin(Alpha) + c.

Each of us learned about what a perimeter is in elementary grades. finding the sides of a square known perimeter problems usually do not arise even for those who graduated from school a long time ago and managed to forget the mathematics course. However, not everyone succeeds in solving a similar problem with respect to a rectangle or a right-angled triangle without a hint.

Instruction

Let's assume that there is a right triangle with sides a, b and c, in which one of the angles is 30 and the second is 60. The figure shows that a = c*sin?, and b = c*cos?. Knowing that the perimeter of any figure, in and a triangle, is equal to the sum all its sides, we get: a + b + c = c * sin ? + c * cos + c = p From this expression, you can find the unknown side c, which is the hypotenuse for the triangle. So how's the angle? = 30, after transformation we get: p/,b=c*cos ?=p*sqrt(3)/

As mentioned above, the diagonal of a rectangle divides it into two right triangles with angles of 30 and 60 degrees. Since p=2(a + b), width a and length b rectangle can be found based on the fact that the diagonal is the hypotenuse of right triangles: a = p-2b/2=p/2
b= p-2a/2=p/2These are two rectangle equations. The length and width of this rectangle are calculated from them, taking into account the resulting angles when drawing its diagonal.

Instruction

1. Perimeter rectangular trapeze equal to the sum of the lengths of the sides of 2 bases and 2 sides. Task 1. Find the perimeter of a rectangular trapeze, if the lengths of all its sides are known. To do this, add all four values: P (perimeter) = a + b + c + d. This is the most primitive version of finding the perimeter, tasks with other initial data, in the final output, are reduced to it. Let's look at the options.

2. Task 2. Find the perimeter of a rectangular trapeze, if the lower base AD = a is known, the lateral side not perpendicular to it is CD = d, and the angle at this lateral side ADC is Alpha. Solution. Draw the height trapeze from the vertex C to a larger base, we get the segment CE, the trapezoid is divided into two figures - the rectangle ABCE and the right triangle ECD. The hypotenuse of a triangle is the lateral side we know trapeze CD, one of the legs is equal to the perpendicular side trapeze(according to the rectangle rule, two parallel sides are equal - AB \u003d CE), and the other is a segment whose length is equal to the difference of the bases trapeze ED=AD-BC.

3. Find the legs of the triangle: using the formulas CE = CD*sin(ADC) and ED = CD*cos(ADC). Now calculate the upper base - BC = AD - ED = a - CD*cos(ADC) = a - d*cos (Alpha). Find out the length of the perpendicular side - AB \u003d CE \u003d d * sin (Alpha). It turns out that you got the lengths of all sides of the rectangular trapeze .

4. Add the resulting values, this will be the perimeter of a rectangular trapeze😛 = AB + BC + CD + AD = d*sin(Alpha) + (a - d*cos(Alpha)) + d + a = 2*a + d*(sin(Alpha) - cos(Alpha) + 1 ).

5. Task 3. Find the perimeter of a rectangular trapeze, if we know the lengths of its bases AD \u003d a, BC \u003d c, the length of the perpendicular side AB \u003d b and the acute angle with a different side ADC \u003d Alpha. Solution. Draw a perpendicular CE, get a rectangle ABCE and a triangle CED. Now find the length of the hypotenuse of the triangle CD \u003d AB / sin (ADC) \u003d b / sin (Alpha). It turns out that you got the lengths of all sides.

6. Add up the resulting values: P = AB + BC + CD + AD = b + c + b/sin(Alpha) + a = a + b*(1+1/sin(Alpha) + c.

Each of us learned about what a perimeter is in elementary grades. finding the sides of a square with a known perimeter of problems usually does not appear even for those who graduated from school a long time ago and managed to forget the mathematics course. However, not everyone can solve a similar problem with respect to a rectangle or a right-angled triangle without a hint.

Instruction

1. How to solve a problem in geometry, in the condition of which only the perimeter and angles are given? Of course, if we are talking about an acute triangle or a polygon, then it is unrealistic to solve such a problem without knowing the length of one of the sides. However, if we are talking about a right-angled triangle or rectangle, then along a given perimeter it is possible to detect its sides. The rectangle has length and width. If we draw a diagonal of a rectangle, we can find that it divides the rectangle into two right-angled triangles. The diagonal is the hypotenuse, and the length and width are the legs of these triangles. For a square, which is a special case of a rectangle, the diagonal is the hypotenuse of a right isosceles triangle.

2. Imagine that there is a right triangle with sides a, b and c, in which one of the angles is 30, and the second is 60. The figure shows that a = c*sin?, and b = c*cos?. Knowing that the perimeter of any figure, including a triangle, is equal to the sum of all its sides, we get: a + b + c = c * sin ? + c * cos + c = p for a triangle. Because the corner? = 30, after reforming we get: p/,b=c*cos ?=p*sqrt(3)/

3. As mentioned above, the diagonal of a rectangle divides it into two right-angled triangles with angles of 30 and 60 degrees. Because the perimeter of the rectangle is p=2(a + b), width a and length b rectangles can be detected based on the fact that the diagonal is the hypotenuse of right triangles: a = p-2b/2=p/2b= p-2a/2=p/2These two equations are expressed in terms of the perimeter of the rectangle. The length and width of this rectangle are calculated from them, taking into account the resulting angles when drawing its diagonal.

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Note!
How to find the length of a rectangle if the perimeter and width are known? Subtract twice the width from the perimeter to get twice the length. Then we divide it in half in order to find the length.

Useful advice
Even from the original school, many remember how to find the perimeter of any geometric figure: it is enough to find out the length of all its sides and find their sum. It is known that in such a figure as a rectangle, the lengths of the sides are equal in pairs. If the width and height of a rectangle are the same length, then it is called a square. Usually, the length of a rectangle is called the largest of the sides, and the width is the smallest.

Perimeter(P) - the sum of the lengths of all sides of the figure, and the quadrangle has four of them. This means that in order to find the perimeter of a quadrilateral, it is necessary to easily add the lengths of all its sides. But we know such figures as a rectangle, a square, a rhombus, that is, positive quadrangles. Their perimeters are determined by special methods.

Instruction

1. If this figure is a rectangle (or parallelogram) ABCD, then it has the following properties: parallel sides are pairwise equal (see figure). AB = SD and AC = VD. Knowing such a ratio of the sides in this figure, it is possible to derive the perimeter rectangle(and parallelogram): P \u003d AB + SD + AC + VD. Let some sides be equal to the number a, the other to the number b, then P \u003d a + a + b + b \u003d 2 * a \u003d 2 * b \u003d 2 * (a + c). Example 1. In a rectangle ABCD, the sides are AB = CD = 7 cm and AC = VD = 3 cm. Find the perimeter of such a rectangle. Solution: P \u003d 2 * (a + c). P \u003d 2 * (7 +3) \u003d 20 cm.

2. When solving problems for the sum of the lengths of the sides with a figure called a square or a rhombus, you should use a slightly modified perimeter formula. A square and a rhombus are figures that have identical four sides. Based on the definition of the perimeter, P \u003d AB + SD + AC + VD and allowing the length to be denoted by the letter a, then P \u003d a + a + a + a \u003d 4 * a. Example 2. A rhombus has a side length of 2 cm. Find its perimeter. Solution: 4*2 cm = 8 cm.

3. If the given quadrilateral is a trapezoid, then in this case it is easy to add the lengths of its four sides. P \u003d AB + SD + AC + VD. Example 3. Find the perimeter of the trapezoid ABCD if its sides are equal: AB = 1 cm, SD = 3 cm, AC = 4 cm, ID = 2 cm. Solution: P = AB + CD + AC + ID = 1 cm + 3 cm + 4 cm + 2 cm = 10 cm. It may happen that the trapezoid is equilateral (it has two sides equal), then its perimeter can be reduced to the formula: P \u003d AB + SD + AC + VD \u003d a + b + a + c \u003d 2 * a + c + c. Example 4. Find the perimeter of an isosceles trapezoid if its side faces are 4 cm, and the bases are 2 cm and 6 cm. Solution: P \u003d 2 * a + b + c \u003d 2 * 4 cm + 2 cm + 6 cm \u003d 16 cm.

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Useful advice
Nobody bothers to find the perimeter of a quadrilateral (and any other figure) as the sum of the lengths of the sides, without applying the derived formulas. They are given for comfort and ease of calculation. The method of solution is not a mistake, the correct result and the ability to use mathematical terminology are significant.

Tip 4: How to find the bases of a rectangular trapezoid

Instruction

1. Task 1. Find the bases BC and AD of a rectangular trapeze, if we know the length of the diagonal AC = f; the length of the lateral side CD = c and the angle with it ADC = ?. Solution: Look at the right triangle CED. The hypotenuse c and the angle between the hypotenuse and the leg EDC are famous. Find the lengths of the sides CE and ED: using the angle formula CE = CD*sin(ADC); ED=CD*cos(ADC). It turns out: CE = c*sin?; ED=c*cos?.

2. Consider right triangle ACE. You know the hypotenuse AC and the leg CE, find the side AE according to the rule of a right triangle: the sum of the squares of the legs is equal to the square of the hypotenuse. It turns out: AE(2) = AC(2) - CE(2) = f(2) - c*sin?. Calculate the square root of the right side of the equation. You have found the upper base of the rectangular trapeze .

3. The length of the base AD is the sum of the lengths of the 2 segments AE and ED. AE = square root(f(2) - c*sin?); ED = c*cos?). It turns out: AD = square root(f(2) - c*sin?) + c*cos?. Have you found the lower base of the rectangular trapeze .

4. Task 2. Find the bases BC and AD of a rectangular trapeze, if we know the length of the diagonal BD = f; the length of the lateral side CD = c and the angle with it ADC = ?. Solution: Look at the right triangle CED. Find the lengths of the sides CE and ED: CE = CD*sin(ADC) = c*sin?; ED = CD*cos(ADC) = c*cos?.

5. Consider the rectangle ABCE. According to the property of the rectangle AB = CE = c*sin?. Look at the right triangle ABD. According to the property of a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Therefore AD(2) = BD(2) - AB(2) = f(2) - c*sin?. You have found the lower base of the rectangular trapeze AD = square root(f(2) - c*sin?).

6. By the rectangle rule BC = AE = AD - ED = square root(f(2) - c*sin?) - c*cos?. Have you found the upper base of the rectangular trapeze .

A trapezoid is a quadrilateral with two parallel and two non-parallel sides. In order to calculate its perimeter, you need to know the dimensions of all sides of the trapezoid. In this case, the data in the tasks may be different.

You will need

- calculator;
- tables of sines, cosines and tangents;
- paper;
- drawing accessories.

Instruction

1. The most primitive version of the problem is when all sides of a trapezoid are given. In this case, they should be easily folded. It is allowed to use the following formula: p=a+b+c+d, where p is the perimeter, and the letters a, b, c and d indicate the sides opposite the corners indicated by the corresponding capital letters.

2. There is an isosceles trapezoid, it is enough to fold its two bases and add to them twice the size of the side. That is, the perimeter in this case is calculated by the formula: p \u003d a + c + 2b, where b is the side of the trapezoid, and and c are the bases.

3. The calculations will be somewhat longer if one of the sides needs to be calculated. Let's say we know the long base, the angles adjacent to it and the height. You need to calculate the short base and side. To do this, draw a trapezoid ABCD, draw a height BE from the upper corner B. You will get a triangle ABE. You are given angle A, respectively, you know its sine. The problem data also contains the height BE, which at the same time is the leg of a right triangle opposite the angle you know. In order to find the hypotenuse AB, which is at the same time the side of the trapezoid, it is enough to divide BE by sinA. Correctly also find the length of the 2nd side. To do this, you need to draw a height from a different upper corner, that is, CF. Now you know the greater base and sides. To calculate the perimeter, this is not much, you also need the size of a smaller base. Accordingly, in 2 triangles formed inside the trapezoid, it is necessary to find the sizes of the segments AE and DF. This can be done, say, through the cosines of the angles A and D known to you. The cosine is the ratio of the adjacent leg to the hypotenuse. In order to find the leg, it is necessary to multiply the hypotenuse by the cosine. Next, calculate the perimeter using the same formula as in the first step, that is, adding up all the sides.

4. Another option: given two bases, a height and one of the sides, you need to find the second side. This is also better done using trigonometric functions. To do this, draw a trapezoid. It is possible, you know the bases AD and BC, as well as the side AB and the height BF. From these data, you can find angle A (through the sine, that is, the ratio of the height to the famous side), segment AF (through the cosine or tangent, from the fact that the angle is more familiar to you. Recall also the properties of the angles of a trapezoid - the sum of the angles adjacent to one side , is 180°. Draw height CF. You have another right-angled triangle in which you need to find the hypotenuse CD and leg DF. Start with the leg. Subtract from the length of the lower base the length of the upper, and from the resulting total - the length of the segment that is more closely known to you AF Now in a right-angled triangle CFD you know two legs, that is, you can find the tangent of angle D, and from it the angle itself.Later on, it will remain to calculate the side CD through the sine of the same angle, as was already described above.

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A trapezoid is a quadrangular geometric figure that has two parallel sides, called bases, and two non-parallel sides. If the sides are equal, then the figure is called an isosceles trapezoid. Rectangular trapezoid - when one side forms a right angle with the base. To find the perimeter of a trapezoid, you can use one of the methods, depending on the source data.

How to find the perimeter of a trapezoid when the length of the sides and bases is known

In this case, there are no difficulties. Using the formula P=a+b+c+d and substituting all the known data, we can easily find the perimeter of the trapezoid. For example: a=5, b=4, c=6, d=4. Using the formula, we get P=5+4+6+4=19

This method cannot be used if the length of at least one of the sides is not known.

How to find the perimeter of a trapezoid when the length of the sides, top base and height are known

Divide the trapezoid into two triangles and a rectangle.

In order to be able to use the formula P=a+b+c+d, it is necessary to find the lower base. It can be represented as an expression k+a+n.

Next, we use the Pythagorean theorem. Let's write the formula for the first triangle c^2=h^2+k^2. After transformations we get k=(c^2-h^2)^1/2. For the second triangle: b^2=h^2+n^2, total n=(b^2-h^2)^1/2. After all the calculations, we get P=a+b+(n+a+k)+c.

How to find the perimeter of a trapezoid when both bases and height are known (for an isosceles trapezoid)

As in the previous method, you need to divide the trapezoid into a rectangle and two triangles. The hypotenuses of the triangles are also the sides of the trapezoid that need to be found. The smaller leg is found as follows.

Since the trapezoid is isosceles, subtract the length of the smaller base from the length of the larger base and divide in half, i.e. d1=d2=(d-a)/2.

Using the Pythagorean theorem, we find the sides c=(d(1)^2+h^2)^1/2. Next, using the formula P=a+2c+d, we calculate the perimeter.

How to find the perimeter of a trapezoid when the bottom base, sides and bottom corners are known

Consider an example where the bottom base AD, the sides AB and CD, and the angles BAD and CDA are known.

From vertices B and C we draw two heights, which form a rectangle and two right-angled triangles. In triangle ABK, side AB is the hypotenuse. It remains to find the legs using the formula BK=AB*sin(BAK) and AK=AB*cos(BAK). Since BK and CN are heights, they are equal. Using the same formula, we find ND=CD*cos(CDN). It remains to calculate BC=AD-AK-ND. Now you need to fold all the sides and the answer is ready.

How to find the perimeter of a trapezoid when the length of the sides and midline is known

The midline of a trapezoid is equal to half the sum of the lengths of its bases, i.e. f=(a+d)/2. When the length of the bases is unknown, but the dimensions of the sides and midline are given, the perimeter is found by the formula P=2*f+c+b.

As you can see, finding the perimeter of a trapezoid is not so difficult. Starting to solve the problem, you only need to determine what quantities are known and what method can be used. And then it will not be difficult to solve even a complex problem.

A trapezoid is a two-dimensional geometric figure that has four vertices and only two parallel sides. If the length of 2 of its non-parallel sides is identical, then the trapezoid is called isosceles or isosceles. The boundary of such a polygon, composed of its sides, is usually denoted Greek word"perimeter". Depending on the set of initial data, it is necessary to calculate the length of the perimeter using various formulas.

Instruction

1. If the lengths of both bases (a and b) and the length of the lateral side (c) are known, then the perimeter (P) of this geometric figure is calculated very primitively. Because the trapezoid is isosceles, its sides have identical lengths, which means that you know the lengths of all sides - add them primitively: P = a + b + 2 * c.

2. If the lengths of both bases of the trapezoid are unfamiliar, but the lengths of the midline (l) and lateral side (c) are given, then these data are sufficient to calculate the perimeter (P). The median line is parallel to both bases and equal in length to their half-sum. Double this value and add to it also twice the length of the lateral side - this will be the perimeter of an isosceles trapezoid: P = 2*l+2*c.

3. If the lengths of both bases (a and b) and the height (h) of an isosceles trapezoid are known from the conditions of the problem, then with the help of these data it is possible to restore the length of the missing side. This can be done by looking at a right triangle, in which the hypotenuse will be an unfamiliar side, and the legs will be the height and a short segment, the one that it cuts off from the long base of the trapezoid. The length of this segment can be calculated by dividing in half the difference between the lengths of the larger and smaller bases: (a-b) / 2. The length of the hypotenuse (lateral side of the trapezoid), according to the Pythagorean theorem, will be equal to square root from the sum of the squared lengths of both driven legs. Replace the side length in the formula from the first step with the resulting expression, and you will get the following perimeter formula: P \u003d a + b + 2 *? (h? + (a-b)? / 4).

4. If in the conditions of the problem the lengths of the smaller base (b) and the side (c), as well as the height of the isosceles trapezoid (h), are given, then considering the same auxiliary triangle as in the previous step, you will have to calculate the length of the leg. Again, use the Pythagorean theorem - the desired value will be equal to the root of the difference between the squared length of the side (hypotenuse) and the height (leg):? (c? -h?). According to this segment of the unfamiliar base of the trapezoid, it is possible to restore its length - double this expression and add the length of the short base to the total: b + 2 *? (c? -h?). Substitute this expression into the formula from the first step and find the perimeter of an isosceles trapezoid: P = b+2*?(c?-h?)+b+2*c = 2*(?(c?-h?)+b+c ).

Tip 2: How to find the sides of an isosceles trapezoid

A trapezoid is a quadrilateral with two parallel sides. These sides are called bases. Their final points are united by segments, which are called sides. An isosceles trapezoid has equal sides.

You will need

- isosceles trapezoid;
are the lengths of the bases of the trapezoid;
- the height of the trapezoid;
- paper;
- pencil;
- ruler.

Instruction

1. Construct a trapezoid according to the conditions of the problem. You must be given several parameters. As usual, these are both bases and height. But other data are also acceptable - one of the bases, its inclination to it of the lateral side and height. Designate the trapezoid as ABCD, let the bases be a and b, designate the height as h, and the sides as x. Since the trapezoid is isosceles, its sides are equal.

2. From vertices B and C draw heights to the lower base. Designate the intersection points as M and N. You get two right-angled triangles - AMB and CND. They are equal because, according to the conditions of the problem, their hypotenuses AB and CD, as well as the legs BM and CN, are equal. Accordingly, segments AM and DN are also equal to each other. Designate their length as y.

3. In order to find the length of the sum of these segments, it is necessary to subtract the length of the base b from the length of the base a. 2y=a-b. Accordingly, one such segment will be equal to the difference of the bases divided by 2. y=(a-b)/2.

4. Find the length of the lateral side of the trapezoid, which at the same time is the hypotenuse of a right triangle with the legs you know. Calculate it using the Pythagorean theorem. It will be equal to the square root of the sum of the squares of the height and the difference of the bases, divided by 2. That is, x=?y2+h2=?(a-b)2/4+h2.

5. Knowing the height and angle of inclination of the side to the base, make the same constructions. In this case, the base difference does not need to be calculated. Use the sine theorem. The hypotenuse is equal to the length of the leg multiplied by the sine of the opposite angle. In this case x=h*sinCDN or x=h*sinBAM.

6. If you are given the angle of inclination of the side of the trapezoid not to the lower, but to the upper base, find the required angle based on the property of parallel lines. Recall one of the properties of an isosceles trapezoid, according to which the angles between one of the bases and the sides are equal.

Note!
Review the properties of an isosceles trapezoid. If you divide both of its bases in half and draw a line through these points, then it will be the axis of this geometric figure. If you lower the height from one vertex of the upper base to the lower one, then two segments will be obtained on this latter. Let's say, in this case, these are segments AM and DM. One of them is equal to half the sum of the bases a and b, and the other is half their difference.

Tip 3: How to discover middle line isosceles trapezium

A trapezoid is a quadrilateral that has only two parallel sides - they are called the bases of this figure. If at the same time the lengths of the other 2 - lateral - sides are identical, the trapezoid is called isosceles or isosceles. The line that connects the midpoints of the sides is called the midline of the trapezoid and can be calculated in several ways.

Instruction

1. If the lengths of both bases (A and B) are known, to calculate the length of the midline (L) use the main quality of this element of an isosceles trapezoid - it is equal to half the sum of the lengths of the bases: L \u003d? * (A + B). Say, in a trapezoid with bases having lengths of 10cm and 20cm, the middle line should be equal to? * (10 + 20) = 15cm.

2. The middle line (L) together with the height (h) of an isosceles trapezoid is a factor in the formula for calculating the area (S) of this figure. If these two parameters are given in the initial conditions of the problem, to calculate the length of the midline, divide the area by the height: L = S/h. Say, with an area of 75 cm? an isosceles trapezoid 15 cm high should have a median line 75/15 = 5 cm long.

3. With a known perimeter (P) and the length of the lateral side (C) of an isosceles trapezoid, it is also not difficult to calculate the midline (L) of the figure. Subtract two lengths of the sides from the perimeter, and the remaining value will be the sum of the lengths of the bases - divide it in half, and the problem will be solved: L \u003d (P-2 * C) / 2. Say, with a perimeter of 150 cm and a side length of 25 cm, the length of the midline should be (150-2 * 25) / 2 = 50 cm.

4. Knowing the lengths of the perimeter (P) and height (h), as well as the value of one of the acute angles (?) of an isosceles trapezoid, it is also possible to calculate the length of its midline (L). In a triangle composed of height, side and part of the base, one of the angles is right, and the value of the other is known. This will allow you to calculate the length of the side using the sine theorem - divide the height by the sine of the known angle: h / sin (?). After that, substitute this expression into the formula from the previous step and you will get the following equality: L = (P-2*h/sin(?))/2 = P/2-h/sin(?). Say if the lead angle is 30°, the height is 10cm, and the perimeter is 150cm, the length of the midline should be calculated as follows: 150/2-10/sin(30°) = 75-20 = 55cm.

Tip 4: How to Find the Perimeter of an Isosceles Triangle

Perimeter is the sum of all sides of a polygon. In regular polygons, the well-defined connectivity between the sides makes it easier to find the perimeter.

Instruction

1. In an arbitrary figure bounded by various segments of a polyline, the perimeter is determined by successive measurements of the sides and summation of the results of the measurement. For positive polygons, finding the perimeter is permissible by calculating formulas that consider the relationships between the sides of the figure.

2. In an arbitrary triangle with sides a, b, c, the perimeter P is calculated by the formula: P \u003d a + b + c. An isosceles triangle has two sides equal to each other: a \u003d b, and the formula for finding the perimeter is simplified to P \u003d 2 * a + c.

3. If in an isosceles triangle the dimensions of not all sides are given by condition, then to find the perimeter it is allowed to use other known parameters, say the area of the triangle, its angles, heights, bisectors and medians. Let's say if only two are famous equal sides an isosceles triangle and any of its angles, then find the third side using the sine theorem, from which it follows that the ratio of the side of the triangle to the sine of the opposite angle is a continuous value for this triangle. Then the unknown side can be expressed through the famous side: a=b*SinA/SinB, where A is the angle opposite the unknown side a, B is the angle contrary to the famous side b.

4. If the area S of an isosceles triangle and its base b are known, then from the formula for determining the area of \u200b\u200bthe triangle S \u003d b * h / 2, find the height h: h \u003d 2 * S / b. This height, lowered to the base b, divides the given isosceles triangle into two equal right triangles. The sides a of the initial isosceles triangle are the hypotenuses of the right triangles. By the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs b and h. Then the perimeter P of an isosceles triangle is calculated by the formula: P=b+2*?(b?/4) +4*S?/b?).

Tip 5: How to find the base of an isosceles trapezoid

A trapezoid is a quadrilateral whose bases lie on 2 parallel lines, while the other two sides are not parallel. Finding the base of an isosceles trapezoid is required both when passing the theory and solving problems in educational institutions, and in a number of professions (engineering, architecture, design).

Instruction

1. An isosceles (or isosceles) trapezoid has non-parallel sides, as well as the angles that form when crossing the lower base, are equal.

2. The trapezoid has two bases, and in order to find them, you must first identify the figure. Let an isosceles trapezoid ABCD with bases AD and BC be given. In this case, all parameters are known, except for the bases. Lateral side AB=CD=a, height BH=h and area equal to S.

3. To solve the problem of the base of a trapezoid, it will be easier for everyone to compose a system of equations in order to find the necessary bases through interrelated quantities.

4. Designate the segment BC as x, and AD as y, so that in the future it will be comfortable to handle formulas and understand them. If you do not do this right away, you can get confused.

5. Write out all the formulas that will fit in solving the problem, using the famous data. The formula for the area of an isosceles trapezoid: S=((AD+BC)*h)/2. Pythagorean theorem: a*a = h*h +AH*AH .

6. Recall the quality of an isosceles trapezoid: the heights emerging from the top of the trapezoid cut off equal segments on a large base. From here it follows that two bases can be connected according to the formula following from this property: AD=BC+2AH or y=x+2AH

7. Locate leg AH by following the Pythagorean theorem, which you wrote down earlier. Let it be equal to some number k. Then the formula following from the property of an isosceles trapezoid will look like this: y=x+2k.

8. Express the unknown quantity in terms of the area of the trapezoid. You should get: AD=2*S/h-BC or y=2*S/h-x.

9. Later, substitute these numerical values into the resulting system of equations and solve it. The solution of any system of equations can be found mechanically in the MathCAD program.

Useful advice
Be diligent invariably when solving problems to simplify notations and formulas as much as possible. So the decision will be found much faster.

You will need

- calculator;
- tables of sines, cosines and tangents;
- paper;
- drawing accessories.

Instruction

1. The most primitive version of the problem is when all sides of a trapezoid are given. In this case, they must be primitively folded. It is allowed to use the following formula: p=a+b+c+d, where p is the perimeter, and the letters a, b, c and d indicate the sides opposite the corners indicated by the corresponding capital letters.

3. The calculations will be somewhat longer if one of the sides needs to be calculated. Say, the long base, the angles adjacent to it and the height are famous. You need to calculate the short base and side. To do this, draw a trapezoid ABCD, draw a height BE from the upper corner B. You will get a triangle ABE. You know the angle A, respectively, you know its sine. The problem data also contains the height BE, which at the same time is the leg of a right triangle opposite the angle you know. In order to find the hypotenuse AB, which is at the same time the side of the trapezoid, it is enough to divide BE by sinA. Correctly also find the length of the 2nd side. To do this, you need to draw a height from a different upper corner, that is, CF. You now know the larger base and sides. To calculate the perimeter, this is not much, you also need the size of a smaller base. Accordingly, in 2 triangles formed inside the trapezoid, it is necessary to find the sizes of the segments AE and DF. This can be done, say, through the cosines of the angles A and D known to you. The cosine is the ratio of the adjacent leg to the hypotenuse. To find the leg, you need to multiply the hypotenuse by the cosine. Next, calculate the perimeter using the same formula as in the first step, that is, adding up all the sides.

4. Another option: given two bases, a height and one of the sides, you need to find the second side. This is also better done using trigonometric functions. To do this, draw a trapezoid. Perhaps you know the bases AD and BC, as well as the side AB and the height BF. According to these data, you can find the angle A (through the sine, that is, the ratio of the height to the famous side), segment AF (through the cosine or tangent, from the fact that the angle is closer to you. Recall also the properties of the angles of a trapezoid - the sum of the angles adjacent to one side , is 180°. Draw height CF. You have another right-angled triangle in which you need to find the hypotenuse CD and leg DF. Start with the leg. Subtract from the length of the lower base the length of the upper, and from the resulting total - the length of the segment that is more closely known to you AF Now in a right-angled triangle CFD you know two legs, that is, you can find the tangent of angle D, and from it the angle itself.Later on, it remains to calculate the side CD through the sine of the same angle, as described above.

Related videos

Find the perimeter of the trapezoid. Hello! In this publication, we will consider the solution of typical problems included in the exam in mathematics. It is required to calculate the perimeter of a trapezoid. We can say that these are tasks for oral calculations, they are simple. Before deciding, I recommend that you look at the article "". Consider the tasks:

27834. In an isosceles trapezoid, the bases are 12 and 27, the acute angle is 60 0 . Find its perimeter.

In order to find the perimeter, we need to calculate the side. From the vertices of the smaller base we lower the heights:

AD is the hypotenuse in right triangle ADF. We can calculate it using the definition of cosine:

AF we can compute:

Consequently:

Thus the perimeter is 12+27+15+15=69.

*When solving the problem, it was also possible to use the property of the leg lying against the angle of 30°. Look:

∠ADF is equal to 30°, leg AF is equal to half of the hypotenuse AD. AF=7.5 hence AD will be equal to 15.

27835. A straight line drawn parallel to the side of the trapezoid through the end of the smaller base, equal to 4, cuts off a triangle whose perimeter is 15. Find the perimeter of the trapezoid.

The solution is clear! Let's look at the sketch: AD and AE are part of the perimeter, DE=CB are the opposite sides of the parallelogram. That is

It remains to add DC and EB. The condition says DC=4. Since DC and EB are opposite sides of the parallelogram, they are equal:

Thus the perimeter is 15+4+4=23.

That's all, good luck to you!

Sincerely, Alexander Krutitskikh.

Instruction

Instruction

Instruction

Tip 4: How to find the bases of a rectangular trapezoid

Instruction

Instruction

How to find the perimeter of a trapezoid when the length of the sides and bases is known

How to find the perimeter of a trapezoid when the length of the sides, top base and height are known

How to find the perimeter of a trapezoid when both bases and height are known (for an isosceles trapezoid)

How to find the perimeter of a trapezoid when the bottom base, sides and bottom corners are known

How to find the perimeter of a trapezoid when the length of the sides and midline is known

Instruction

Tip 2: How to find the sides of an isosceles trapezoid

Instruction

Tip 3: How to discover middle line isosceles trapezium

Instruction

Tip 4: How to Find the Perimeter of an Isosceles Triangle

Instruction

Tip 5: How to find the base of an isosceles trapezoid

Instruction

Instruction

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