Equal to the sine of an acute angle of a trapezoid. Angles of an isosceles trapezoid

To the simple question “How to find the height of a trapezoid?” There are several answers, all because different starting values ​​can be given. Therefore, the formulas will differ.

These formulas can be memorized, but they are not difficult to derive. You just need to apply previously learned theorems.

Notations used in formulas

In all the below mathematical notations These readings of the letters are correct.

In the source data: all sides

To find the height of a trapezoid in general case you will need to use the following formula:

n = √(c 2 - (((a - c) 2 + c 2 - d 2)/(2(a - c))) 2). Number 1.

Not the shortest, but also found quite rarely in problems. Usually you can use other data.

Formula that will tell you how to find the height isosceles trapezoid in the same situation, much shorter:

n = √(c 2 - (a - c) 2 /4). Number 2.

The problem gives: lateral sides and angles at the lower base

It is assumed that the angle α is adjacent to the side with the designation “c”, respectively, the angle β is to the side d. Then the formula for how to find the height of a trapezoid will be in general form:

n = c * sin α = d * sin β. Number 3.

If the figure is isosceles, then you can use this option:

n = c * sin α= ((a - b) / 2) * tan α. Number 4.

Known: diagonals and angles between them

Typically, these data are accompanied by other known quantities. For example, the bases or the middle line. If the reasons are given, then to answer the question of how to find the height of a trapezoid, the following formula will be useful:

n = (d 1 * d 2 * sin γ) / (a ​​+ b) or n = (d 1 * d 2 * sin δ) / (a ​​+ b). Number 5.

It is for general view figures. If an isosceles is given, then the notation will change like this:

n = (d 1 2 * sin γ) / (a ​​+ b) or n = (d 1 2 * sin δ) / (a ​​+ b). Number 6.

When in a task we're talking about O midline trapezoid, then the formulas for finding its height become:

n = (d 1 * d 2 * sin γ) / 2m or n = (d 1 * d 2 * sin δ) / 2m. Number 5a.

n = (d 1 2 * sin γ) / 2m or n = (d 1 2 * sin δ) / 2m. Number 6a.

Among the known quantities: area with bases or midline

These are perhaps the shortest and simple formulas how to find the height of a trapezoid. For an arbitrary figure it will be like this:

n = 2S / (a ​​+ b). Number 7.

It’s the same, but with a known middle line:

n = S/m. Number 7a.

Oddly enough, for an isosceles trapezoid the formulas will look the same.

Tasks

No. 1. To determine the angles at the lower base of the trapezoid.

Condition. Given an isosceles trapezoid, side which is 5 cm. Its bases are 6 and 12 cm. You need to find the sine acute angle.

Solution. For convenience, you should enter a designation. Let the lower left vertex be A, all the rest in a clockwise direction: B, C, D. Thus, the lower base will be designated AD, the upper one - BC.

It is necessary to draw heights from vertices B and C. The points that indicate the ends of the heights will be designated H 1 and H 2, respectively. Since all the angles in the figure BCH 1 H 2 are right angles, it is a rectangle. This means that the segment H 1 H 2 is 6 cm.

Now we need to consider two triangles. They are equal because they are rectangular with the same hypotenuses and vertical legs. It follows from this that their smaller legs are equal. Therefore, they can be defined as the quotient of the difference. The latter is obtained by subtracting the upper one from the lower base. It will be divided by 2. That is, 12 - 6 must be divided by 2. AN 1 = N 2 D = 3 (cm).

Now from the Pythagorean theorem you need to find the height of the trapezoid. It is necessary to find the sine of an angle. VN 1 = √(5 2 - 3 2) = 4 (cm).

Using the knowledge of how the sine of an acute angle is found in a triangle with a right angle, we can write the following expression: sin α = ВН 1 / AB = 0.8.

Answer. The required sine is 0.8.

No. 2. To find the height of a trapezoid using a known tangent.

Condition. For an isosceles trapezoid, you need to calculate the height. It is known that its bases are 15 and 28 cm. The tangent of the acute angle is given: 11/13.

Solution. The designation of vertices is the same as in previous task. Again you need to draw two heights from upper corners. By analogy with the solution to the first problem, you need to find AN 1 = N 2 D, which is defined as the difference of 28 and 15 divided by two. After calculations it turns out: 6.5 cm.

Since the tangent is the ratio of two legs, we can write the following equality: tan α = AH 1 / VN 1 . Moreover, this ratio is equal to 11/13 (according to the condition). Since AN 1 is known, the height can be calculated: BH 1 = (11 * 6.5) / 13. Simple calculations give a result of 5.5 cm.

Answer. The required height is 5.5 cm.

No. 3. To calculate the height using known diagonals.

Condition. It is known about the trapezoid that its diagonals are 13 and 3 cm. You need to find out its height if the sum of the bases is 14 cm.

Solution. Let the designation of the figure be the same as before. Let's assume that AC is the smaller diagonal. From vertex C you need to draw the desired height and designate it CH.

Now you need to do additional construction. From angle C you need to draw a straight line parallel larger diagonal and find the point of its intersection with the continuation of side AD. This will be D 1. The result is a new trapezoid, inside which a triangle ASD 1 is drawn. This is what is needed to further solve the problem.

The desired height will also be in the triangle. Therefore, you can use the formulas studied in another topic. The height of a triangle is defined as the product of the number 2 and the area divided by the side to which it is drawn. And the side turns out to be equal to the sum of the bases of the original trapezoid. This comes from the rule by which the additional construction was made.

In the triangle under consideration, all sides are known. For convenience, we introduce the notation x = 3 cm, y = 13 cm, z = 14 cm.

Now you can calculate the area using Heron's theorem. The semi-perimeter will be equal to p = (x + y + z) / 2 = (3 + 13 + 14) / 2 = 15 (cm). Then the formula for the area after substituting the values ​​will look like this: S = √(15 * (15 - 3) * (15 - 13) * (15 - 14)) = 6 √10 (cm 2).

Answer. The height is 6√10 / 7 cm.

No. 4. To find the height on the sides.

Condition. Given a trapezoid, three sides of which are 10 cm, and the fourth is 24 cm. You need to find out its height.

Solution. Since the figure is isosceles, you will need formula number 2. You just need to substitute all the values ​​​​into it and count. It will look like this:

n = √(10 2 - (10 - 24) 2 /4) = √51 (cm).

Answer. n = √51 cm.

Instructions

If the lengths of both bases (b and c) and the same lateral sides (a) by definition are known, then a right triangle can be used to calculate the value of one of its acute angles (γ). To do this, lower the height from any corner adjacent to the short base. A right triangle will be formed by a height (), a side (hypotenuse) and a segment of the long base between the height and the near side (the second leg). The length of this segment can be found by subtracting the length of the smaller one from the length of the larger base and dividing the result in half: (c-b)/2.

Having received the lengths of two adjacent sides right triangle, proceed to calculating the angle between them. The ratio of the length of the hypotenuse (a) to the length of the leg ((c-b)/2) gives the cosine value of this angle (cos(γ)), ​​and the arccosine function will help convert it into the angle in degrees: γ=arccos(2*a/(c-b )). This way you will get the value of one of the acute angles, and since it is isosceles, the second acute angle will have the same value. The sum of all angles must be 360°, which means that the sum of two angles will be equal to the difference between this and twice the acute angle. Since both obtuse angles will also be the same, to find the value of each of them (α), this difference must be divided in half: α = (360°-2*γ)/2 = 180°-arccos(2*a/(c-b)) . Now you have calculations of all the angles of an isosceles trapezoid given the known lengths of its sides.

If the lengths of the sides of the figure are unknown, but its height (h) is given, then you need to proceed according to the same scheme. In this case, in a right triangle made up of , a side and a short segment of a long base, you will know the lengths of two legs. Their ratio determines the tangent of the angle you need, and this trigonometric function also has its own antipode, which converts the tangent value into the angle value - arctangent. The formulas for acute and obtuse angles transform accordingly: γ=arctg(2*h/(c-b)) and α = 180°-arctg(2*h/(c-b)).

To solve this problem using methods vector algebra, you need to know the following concepts: geometric vector sum and dot product of vectors, and you should also remember the sum property internal corners quadrangle.

You will need

• - paper;
• - pen;
• - ruler.

Instructions

A vector is a directed segment, that is, a quantity that is considered fully specified if its length and direction (angle) to a given axis are given. The position of the vector is no longer limited by anything. Two vectors with lengths and the same direction are considered equal. Therefore, when using coordinates, vectors are represented by radius vectors of the points of its end (the origin is at the origin of coordinates).

By definition: the resulting vector geometric sum vectors is a vector that starts from the beginning of the first and has the end of the second, provided that the end of the first is combined with the beginning of the second. This can be continued further, building a chain of similarly located vectors.
Draw the given ABCD with vectors a, b, c and d in Fig. 1. Obviously, with this arrangement the resulting vector is d=a+ b+c.

Scalar product V in this case more convenient based on vectors a and d. Dot product, denoted by (a, d)= |a||d|cosф1. Here φ1 is the angle between vectors a and d.
Dot product of vectors, given by coordinates, is determined by the following:
(a(ax, ay), d(dx, dy))=axdx+aydy, |a|^2= ax^2+ ay^2, |d|^2= dx^2+ dy^2, then
cos Ф1=(axdx+aydy)/(sqrt(ax^2+ ay^2)sqrt(dx^2+ dy^2)).

Angles of an isosceles trapezoid. Hello! This article will focus on solving problems with trapezoids. This group assignments are part of the exam, the problems are simple. We will calculate the angles of the trapezoid, base and height. Solving a number of problems comes down to solving, as they say: where are we without the Pythagorean theorem?

We will work with an isosceles trapezoid. It has equal sides and angles at the bases. There is an article on the trapezoid on the blog.

Note the small and important nuance, which we will not describe in detail during the process of solving the tasks themselves. Look, if we are given two reasons, then larger base heights lowered to it is divided into three segments - one is equal to smaller base(these are the opposite sides of the rectangle), the other two are equal to each other (these are the legs of equal right triangles):

A simple example: given two bases of an isosceles trapezoid 25 and 65. The larger base is divided into segments as follows:

*And further! Not included in tasks letter designations. This was done deliberately so as not to overload the solution with algebraic refinements. I agree that this is mathematically illiterate, but the goal is to get the point across. And you can always make the designations for vertices and other elements yourself and write down a mathematically correct solution.

Let's consider the tasks:

27439. The bases of an isosceles trapezoid are 51 and 65. The sides are 25. Find the sine of the acute angle of the trapezoid.

In order to find the angle, you need to construct the heights. In the sketch we denote the data in the quantity condition. The lower base is 65, with heights it is divided into segments 7, 51 and 7:

In a right triangle, we know the hypotenuse and leg, we can find the second leg (the height of the trapezoid) and then calculate the sine of the angle.

According to the Pythagorean theorem, the indicated leg is equal to:

Thus:

Answer: 0.96

27440. The bases of an isosceles trapezoid are 43 and 73. The cosine of an acute angle of a trapezoid is 5/7. Find the side.

Let's construct the heights and note the data in the magnitude condition; the lower base is divided into segments 15, 43 and 15:

27441. The greater base of an isosceles trapezoid is 34. The side is 14. The sine of an acute angle is (2√10)/7. Find the smaller base.

Let's build heights. In order to find a smaller base we need to find what equal to the segment being a leg in a right triangle (indicated in blue):

We can calculate the height of the trapezoid and then find the leg:

Using the Pythagorean theorem we calculate the leg:

So the smaller base is:

27442. The bases of an isosceles trapezoid are 7 and 51. The tangent of an acute angle is 5/11. Find the height of the trapezoid.

Let's construct the heights and mark the data in the magnitude condition. The lower base is divided into segments:

What to do? We express the tangent of the angle known to us at the base in a right triangle:

27443. The smaller base of an isosceles trapezoid is 23. The height of the trapezoid is 39. The tangent of an acute angle is 13/8. Find a larger base.

We build the heights and calculate what the leg is equal to:

Thus the larger base will be equal to:

27444. The bases of an isosceles trapezoid are 17 and 87. The height of the trapezoid is 14. Find the tangent of the acute angle.

We build heights and mark known values ​​​​on the sketch. The lower base is divided into segments 35, 17, 35:

By definition of tangent:

77152. The bases of an isosceles trapezoid are 6 and 12. The sine of an acute angle of a trapezoid is 0.8. Find the side.

Let's build a sketch, construct heights and mark known values, the larger base is divided into segments 3, 6 and 3:

Let's express the hypotenuse, designated as x, through the cosine:

From the main trigonometric identity let's find cosα

Thus:

27818. What is equal to larger angle isosceles trapezoid, if it is known that the difference between the opposite angles is 50 0? Give your answer in degrees.

From the geometry course we know that if we have two parallel lines and a transversal, the sum of the internal one-sided angles is equal to 180 0. In our case it is

The condition says that the difference between opposite angles is 50 0, that is

Properties of a rectangular trapezoid

• U rectangular trapezoid and two angles must be right
• Both right angles rectangular trapezoid necessarily belong to adjacent vertices
• Both right angles in a rectangular trapezoid they are necessarily adjacent to the same side
• Diagonals of a rectangular trapezoid form on one of the sides right triangle
• Side Length of a trapezoid perpendicular to the bases is equal to its height
• At a rectangular trapezoid the bases are parallel, one side is perpendicular to the bases, and the second side is inclined to the bases
• At a rectangular trapezoid two angles are right, and the other two are acute and obtuse

Task