In △ABC, AD and BE are the angle bisectors of ∠A and ∠B and DE ║ AB . Find the length of DE if perimeter of ABDE is 30 cm and AB = 12 cm.

Question

Hi i need help within the next `hr please help

 

Answer 1

Let x = DE. Since DE is parallel to AB, by the alternate interior angles theorem, m<BAD = m<ADE and m<ABE = m<DEB. As AD is an angle bisector of <A, m<EAD = m<DAB; since BE is an angle bisector of <B, m<ABE = m<EBD. Therefore, m<EAD = m<ADE and m<EBD = m<BED. The triangles ADE and EDB are then isosceles with AE = ED and ED = DB. So AE = DE = DB = x, and since the perimeter of ABDE is 30 cm, then 12 + 3x = 30, forcing x = 6 cm. That is, the length of DE is 6 cm.