Question

В трапеции ABCD BC параллельно AD диагонали пересекаются в точке О площадь треугольника BOC-3, а площадь треугольника AOD-27 найдите АС Если АО-6

Answer 1

По подобию.
Треугольник ВОС подобен треугольнику АОД.
Вспоминаем отношение.
S треугольника АОД/S треугольника ВОС=k^2
Получаем
27/3=k^2
9=k^2
k= корень из 9 = 3.
Снова по подобию получаем, что 
AO/OC=k
OC=AO/k=6/3=2
AC=AO+OC=6+2=8
Сторона АС=8

Answer 2

Problem Analysis

We are given a trapezoid ABCD, where BC is parallel to AD. The diagonals intersect at point O. The area of triangle BOC is 3, and the area of triangle AOD is 27. We need to find the length of AC, given that AO is 6.

Solution

To find the length of AC, we can use the fact that the area of a triangle is equal to half the product of its base and height. Let's denote the length of AC as x.

The area of triangle BOC is given as 3, so we have:

Area of triangle BOC = (1/2) * BC * OC = 3 Similarly, the area of triangle AOD is given as 27, so we have:

Area of triangle AOD = (1/2) * AD * AO = 27 We are also given that AO is 6. Substituting this value into the equation for the area of triangle AOD, we get:

(1/2) * AD * 6 = 27

Simplifying the equation, we have:

AD = 9 Now, we can substitute the values of AD and AO into the equation for the area of triangle BOC:

(1/2) * BC * OC = 3

Since BC is parallel to AD, we can use the fact that corresponding angles formed by parallel lines are equal. This means that triangle BOC is similar to triangle AOD. Therefore, the ratio of their corresponding sides is equal:

BC / AD = OC / AO

Substituting the values of BC, AD, and AO, we have:

BC / 9 = OC / 6

Simplifying the equation, we get:

BC = (3/2) * OC Now, we can substitute the value of BC into the equation for the area of triangle BOC:

(1/2) * (3/2) * OC * OC = 3

Simplifying the equation, we have:

OC * OC = 4

Taking the square root of both sides, we get:

OC = 2 Finally, we can substitute the values of OC and BC into the equation for the length of AC:

AC = AO + OC = 6 + 2 = 8

Therefore, the length of AC is 8.

Answer

The length of AC is 8.

Explanation

We found the length of AC by using the given areas of triangles BOC and AOD, and the fact that BC is parallel to AD. We used the formula for the area of a triangle and the properties of similar triangles to solve for the length of AC.