Angles of an isosceles trapezoid. Hello! This article will focus on solving problems with a trapezoid. This group of tasks is part of the exam, the tasks are simple. We will calculate the angles of the trapezium, base and height. The solution of 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 a blog article about the trapezoid,.
We note a small and important nuance, which we will not describe in detail in the process of solving the tasks themselves. Look, if we have two bases, then the larger base is divided into three segments by the heights lowered to it - one is equal to the smaller base (these are 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! Letter designations are not entered in the tasks. This is done intentionally so as not to overload the solution with algebraic frills. I agree that this is mathematically illiterate, but the goal is to convey the essence. And you can always make the designations of vertices and other elements yourself and write down a mathematically correct solution.
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 plot heights. On the sketch, we denote the data in the size condition. The lower base is 65, it is divided by heights into segments 7, 51 and 7:
In a right triangle, we know the hypotenuse and the 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 specified 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 build the heights and mark 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 lateral side is 14. The sine of an acute angle is (2√10)/7. Find a smaller base.
Let's build heights. In order to find a smaller base, we need to find what the segment that is the leg in a right-angled triangle (indicated in blue) is equal to:
We can calculate the height of the trapezoid, and then find the leg:
By 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 plot 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 we know 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 bigger base.
We build heights and calculate what the leg is equal to:
So the larger base will be:
27444. The bases of an isosceles trapezoid are 17 and 87. The height of the trapezoid is 14. Find the tangent of an 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 the acute angle of the trapezoid is 0.8. Find the side.
Let's build a sketch, build heights and note the known values, the larger base is divided into segments 3, 6 and 3:
We express the hypotenuse, denoted as x, through the cosine:
From the basic trigonometric identity we find cosα
Thus:
27818. What is the largest angle of an isosceles trapezoid if it is known that the difference between the opposite angles is 50 0 ? Give your answer in degrees.
From the course of geometry, we know that if we have two parallel lines and a secant, that the sum of internal one-sided angles is 180 0 . In our case, this
The condition says that the difference of opposite angles is 50 0 , that is
From points D and C we drop two heights:
As mentioned above, they divide the larger base into three segments: one is equal to the smaller base, the other two are equal to each other.
In this case, they are 3, 9, and 3 (for a total of 15). In addition, we note that right-angled triangles are cut off by heights, and they are isosceles, since the angles at the base are equal to 45 0 . It follows that the height of the trapezoid will be equal to 3.
That's all! Good luck to you!
Sincerely, Alexander.
Angles of an isosceles (isosceles) trapezoid
Task.
Decision.
For a convex n-gon, the sum of the angles is 180°(n-2).
Thus, the sum of the angles of an isosceles (isosceles) trapezoid is:
180 (4 - 2) = 360 degrees.
Based on the properties of an isosceles trapezoid that its angles are pairwise equal, we denote one pair of angles as x. Since one angle is 30 degrees greater than the second, the sum of the angles of an isosceles trapezoid is:
x + (x + 30) + x + (x + 30) = 360
4x + 60 = 360
x = 75
Answer: the angles of an isosceles (isosceles) trapezoid are equal to 75 and 105 degrees in pairs.
Task.
Find the angles of an isosceles trapezoid if one angle is 30 degrees greater than the other.
Decision.
To solve the problem, we use the following theorem:
Isosceles trapezoid
Note. This is part of the course with tasks in geometry (section isosceles trapezoid). If you need to solve a problem in geometry, which is not here - write about it in the forum. To denote the action of extracting a square root in solving problems, the symbol is used√ or sqrt(), with the radical expression indicated in brackets.
Task
The bases of an isosceles (isosceles) trapezoid are 8 and 20 centimeters. The lateral side is 10 cm. Find the area of a trapezoid like this one that has a height of 12 cm.
Decision.
From the vertex B of the trapezoid ABCD we lower the altitude BM to the base AD. From the vertex C to the base AD let's lower the height CN. Since MBCN is a rectangle, then
AD=BC+AM+ND
The triangles resulting from the fact that we lowered from the smaller base of an isosceles trapezoid to a larger two heights are equal. Thus,
AD = BC + AM * 2
AM = (AD - BC) / 2
AM = (20 - 8) / 2 = 6 cm
Thus, in the triangle ABM formed by the height lowered from the smaller base of the trapezoid to the larger one, we know the leg and the hypotenuse. The remaining leg, which is also the height of the trapezoid, we find by the Pythagorean theorem:
BM 2 = AB 2 - AM 2
BM 2 = 102 - 62
BM=8cm
Since the height of the trapezoid ABCD is 8 cm, and the height of a similar trapezoid is 12 cm, then the similarity coefficient will be equal to
k = 12 / 8 = 1.5
Since in such figures all geometric dimensions are proportional to each other with a similarity coefficient, we find the area of a similar trapezoid. The product of the half-sum of the bases of a similar trapezoid and the height is expressed in terms of the known geometric dimensions of the original trapezoid and the similarity coefficient:
Ssub = (AD * k + BC * k) / 2 * (BM * k)
Spod \u003d (20 * 1.5 + 8 * 1.5) / 2 * (8 * 1.5) \u003d (30 + 12) / 2 * 12 \u003d 252 cm 2
Answer: 252 cm2
Task
In an isosceles trapezoid, the larger base is 36 cm, the side is 25 cm, the diagonal is 29 cm. Find the area of the trapezoid.
Decision.
From the vertex B of the trapezoid ABCD we lower the altitude BM to the base AD. For the resulting right triangles ABM and BMD, the following is true:
AB 2 = BM 2 + AM 2
AD 2 = BM 2 + MD 2
Since the height of an isosceles trapezoid is simultaneously equal to
BM 2 = AB 2 - AM 2
BM 2 = AD 2 - MD 2
Thus,
AB 2 - AM 2 = AD 2 - MD 2
25 2 - AM 2 = 29 2 - MD 2
Since AD = AM + MD, then
AM + MD = 36
MD=36-AM
Where
25 2 - AM 2 = 29 2 - (36 - AM) 2
625 - AM 2 = 841 - (36 - AM) 2
625 - AM 2 = 841 - (1296 - 72AM + AM 2)
625 - AM 2 = 72AM - 455 - AM 2
625 = 72AM - 455
AM=15
Where MD = 36 - 15 = 21
Since AM \u003d 15, then the value of the smaller base of an isosceles trapezoid will be equal to 36 - 15 * 2 \u003d 6 cm
We find the height of an isosceles trapezoid using the Pythagorean theorem:
BM 2 = AB 2 - AM 2
BM 2 = 625 - 225
BM=20
The area of an isosceles trapezoid is equal to the product of half the sum of the bases and the height of the trapezoid.
S \u003d 1/2 (36 + 6) * 20 \u003d 420 cm 2.
Answer: 420 cm2.
Isosceles trapezoid (part 2)
Note. This is part of the course with tasks in geometry (section isosceles trapezoid). If you need to solve a problem in geometry, which is not here - write about it in the forum. To denote the action of extracting a square root in solving problems, the symbol √ or sqrt () is used, and the radical expression is indicated in brackets.
Task.
In an isosceles trapezoid ABCD, the smaller base BC = 5 cm, the angle ABC = 135 degrees, the height of the trapezoid is 3 cm. Find the larger base.
Decision.
Let us lower the height BE from the vertex B to the base AD.
As a result, angle ABC is equal to the sum of the degree measures of angles ABE and EBC. Since the bases of the trapezoid are parallel, the angle EBC is 90 degrees. Whence the angle ABE = 135 - 90 = 45 degrees.
Since BE is the altitude, then the triangle ABE is a right triangle. Knowing the angle ABE, we determine that the angle EAB is equal to 180º - 90º - 45º = 45º. Whence it follows that the triangle ABE is isosceles, that is, AE = BE = 3 cm.
Since the trapezoid ABCD is isosceles, the larger base is 5 + 3 + 3 = 11 cm.
Answer: the greater base of an isosceles trapezoid is 11 cm.
Task
Find the midline of an isosceles trapezoid whose diagonal is the bisector of an acute angle, whose side is 5, and one of the bases is 2 times the other.
Decision.
Since the bases of the trapezoid are parallel, angle ADB is equal to angle DBC, as are the interior angles lying across. Since the diagonal is a bisector by the condition, the angles ADB and BDC are equal. Whence it follows that the angles CBD and CDB are equal.
2. In an isosceles triangle ABC with base AC, the lateral side AB is 2, and the height drawn to the base is the root of 3. Find the cosine of angle A.
3. In triangle ABC AC=BC , AB=32 , cosA=4\5. find the height CH
isosceles trapezoid. PLEASE WITH DRAWING AND IN DETAILS
Help me please:)
Lines AM, BN and CO are parallel, DM = MN = NO. Find:
1) the length of the segment DC, if:
a) AB=12; b) BC=9cm; c) AD = m
2) the length of segment AB if:
a) BD=16cm; b) AC=18 cm: c) DC=b
3) the length of segment AC, if:
a) CD=27 cm; b) DC=36cm; c) DB=a
Please need it tomorrow :(
2. draw an arbitrary segment AB, Divide it:
a) into 5 equal parts
b) into 6 equal parts
3. Find the angles of an isosceles trapezoid if its smaller base is equal to the side and half the other base.
The distance from the center of a circle inscribed in a rectangular trapezoid to the ends of the larger lateral side is 6 and 8 cm find the area of the trapezoid. task 3. In a right triangle ABC (angle C \u003d 90 degrees) AB \u003d 10 cm, the radius of the circle inscribed in it is 2 cm. Find the area of \u200b\u200bthis triangle. task 4. The point divides the chord AB into segments 12 and 16 cm. Find the diameter of the circle if the distance from point C to the center of the circle is 8 cm. quadrilateral ABCO if angle AOC=120 degrees. .
1.) In an isosceles triangle ABC, the lateral side AB is twice the length of its base AC, and the perimeter is 30 cm. Find the base AC2.) In triangle ABC, the median BD is the bisector of the triangle. Find the perimeter of triangle ABC if the perimeter of triangle ABD is 16 cm and the median BD is 5 cm.
3.) Determine the type of triangle if one of its sides is 5 cm, and the other is
3cm and perimeter is 7cm.
4.) Segment AK - the height of the isosceles triangle ABC drawn to the base BC. Find angles BAK and BKA if angle BAC=46 degrees.
5.) Triangle ABC is isosceles with base AC. Define angle 2 if angle 1 is 68 degrees.
6.) In the triangle ABC, the median CM is drawn. It is known that CM = MB, MAC angle = 53 degrees, MBC angle = 37 degrees. Find the angle ACB.
7.) Determine the type of triangle, two heights of which lay outside the triangle, and draw a drawing if such a triangle exists.
8.) The median BM of triangle ABC is perpendicular to its bisector AD. Find AB if AC = 12 cm.
At the very beginning, we clarify that a trapezoid is a geometric figure, which is a quadrilateral with two parallel opposite sides. They are called the bases of the trapezoid, and the other two are called its sides. When connecting the central points of the sides, you can get the middle line of the figure. These properties of a trapezoid underlie the calculation of all its other characteristics. In order to calculate the base of a trapezoid (large or small), you can use a lot of different approaches. Everything depends on the completeness of the available information about the geometric object. Most of the tasks have data on other sides and angles of the trapezoid in the condition, which greatly simplifies the task. Often the solution is to drop the height to the base and use the Pythagorean theorem to find the right parameters. The calculation of one of the bases with the available information about the area of the trapezoid and the second base does not present any problems at all. Consider the most common cases with examples.
How to find the base of a rectangular trapezoid
A rectangular trapezoid is a trapezoid in which one of the angles is equal to 90 degrees. This is the simplest of all options for calculating the base. As a rule, the condition of the problem contains data about the second base, and the solution is only to determine the fragment of the base that forms the second corner of the figure with the side. As in the case described above, we consider a separate triangle with a base from the desired fragment. According to the Pythagorean theorem, we calculate this part, add or subtract it from the second base and get the desired parameter.
How to find the base of an isosceles trapezoid
It looks like the situation is with an isosceles trapezoid. This concept is understood as such a trapezoid, whose sides are equal. This figure is absolutely symmetrical about the center, because the pairs of angles in it are equal. This is quite convenient, because, having information about at least one angle, we can easily calculate the parameters of all the others. Since the side parts of the trapezoid are equal to each other, then, as in the previous problem, we must find the base through one small fragment of it. The length of the second fragment will exactly match the length of the first. This is also done through the image of the height forming a triangle. Through the parameters of the angles and one side of this triangle, we can easily get the required part of the larger base.
How to find the smallest base of an isosceles trapezoid
If we know the parameters of the larger base, the sides, then this can be done like this. On a larger basis, we lower the height and write down the two Pythagorean theorems. One will reflect the parameters of a triangle in which the diagonal acts as the hypotenuse, the height as one leg, and the larger base as the other leg without a segment cut off by the height.
The second theorem should be relevant for a triangle, which consists of a hypotenuse - a side, a leg - height and a leg - a segment from a larger base.
We compose a system of these equations and solve it. We find the segment cut off by the height from the greater distance. Subtract the doubled parameters of this segment from the parameters of the larger base and get the length of the smaller base.
