Question

In a parallelogram ABCD the length of the sides AD=8 in, AB=3 in. Angle bisectors of ∠A and ∠D split the opposite side into three segments. Find the length of each of these segments.

Answer 1

Answer with explanation:

A Parallelogram ABCD, in which , AD=8 in, and AB=3 in.

Opposite sides as well aas Opposite angles of Parallelogram are equal.

∠A=∠C, ∠B=∠D,AB=CD, AD=BC

It is given that ,∠A and ∠D is bisected by two line segments which divides Side BC into three segments.

Let, ∠A= ∠1 +∠2,such that, ∠1= ∠2.

And, ∠D=∠3+∠4 , such that, ∠3=∠4.

Sum of interior angles on the same side of transversal is 180°.

∠A+∠D=180°

∠D=180°- ∠A

So,

`\angle 4=90^(\circ)-(\angle A)/(2)\\\\\text{Using Angle sum property of triangle}\\\\\angle 4+\angle C+\angle 5=180^(\circ)\\\\90^(\circ)-(\angle A)/(2)+\angle A+\angle 5=180^(\circ)\\\\\angle 5=90^(\circ)-(\angle A)/(2)\\\\\angle 5=\angle 4`

CN=CD----[If opposite angles of triangle are equal , then side opposite to these angles are equal.]

CD=3 in

So, CN=3 in.

Similarly, AB=BM=3 cm

So, MN=8-BM-CN

=8-3-3

=2 in.