AD is the bisector of the angle BAC. BD = CD = 14 cm. The perimeter of triangle ABD is 36 cm.

AD is the bisector of the angle BAC. BD = CD = 14 cm. The perimeter of triangle ABD is 36 cm. Angles DBA and CDA are equal. Find the length of AD.

Consider triangles CDA and DBA.

∠ACD = 180 ° – ∠CAD – ∠CDA = 180 ° – ∠BAD – ∠DBA = ∠ADB.

The angle ACD of one triangle is equal to the angle ADB of the other triangle.

The other two angles of the triangles are equal by condition.

Side AD – common, sides BD = CD.

We got that these triangles are equal. This means that the other parties will also be equal:

CA = DA, AD = AB.

It turns out that triangle ABD is isosceles.

Base BD = 14, perimeter P = 36.

Find AD:

AD + AB = 36 – 14 = 22;

AD = AB = 1/2 * 22 = 11;

AD = 11.

Answer: AD = 11.