Question

В трапеции АВСД с основаниями АД=13 и ВС=7 точка К середина ВД а луч АК биссектриса угла САД найти длину диагонали АС

Answer 1

Problem Analysis

We are given a trapezoid ABCD with bases AD = 13 and BC = 7. Point K is the midpoint of BD, and line AK is the bisector of angle CAD. We need to find the length of diagonal AC.

Solution

To find the length of diagonal AC, we can use the properties of a trapezoid and the angle bisector theorem.

Let's denote the length of diagonal AC as x.

According to the angle bisector theorem, we have:

AK/CK = AD/CD

Since AK is the bisector of angle CAD, we can write:

AK/CK = AD/CD = AD/BC

Substituting the given values, we have:

AK/CK = 13/7

We also know that point K is the midpoint of BD. Therefore, we can write:

BK = KD = 1/2 * BD

Substituting the given value, we have:

BK = KD = 1/2 * 7 = 7/2

Now, we can use the Pythagorean theorem to find the length of AC. In triangle AKC, we have:

AC^2 = AK^2 + CK^2

Substituting the values we found earlier, we have:

AC^2 = (13/7)^2 + (7/2)^2

Simplifying the equation, we get:

AC^2 = 169/49 + 49/4

Combining the fractions, we have:

AC^2 = (169*4 + 49*49)/(49*4)

Simplifying further, we get:

AC^2 = (676 + 2401)/196

AC^2 = 3077/196

Taking the square root of both sides, we find:

AC = sqrt(3077/196)

Calculating the value, we get:

AC ≈ 7.01

Therefore, the length of diagonal AC is approximately 7.01 units.

Answer

The length of diagonal AC in trapezoid ABCD is approximately 7.01 units.