Question

ABCDE is a convex pentagon. Given AB = 5, BC = 12, AE = 13, DE = 8, CD = 6, and m∠B=m∠D=90°. Find the area of ABCDE.

Answer 1

The diagonals AC and EC of the pentagon forms two right triangles and

on isosceles triangle, together which gives the area of the pentagon.

Correct response:

• The area of the convex pentagon is 104 square units.

Methods used to finding the area of a pentagon

The given parameters are;

AB = 5

BC = 12

AE = 13

DE = 8

CD = 6

m∠B = m∠D = 90°

Required:

The area of the convex pentagon ABCDE

Solution:

The area of pentagon ABCDE = Right ΔABC + Right ΔCDE + ΔACE

Area of right triangle ΔABC = $\frac{1}{2}$ × 5 × 12 = 30

Area of right triangle ΔCDE = $\frac{1}{2}$ × 6 × 8 = 24

Length of AC = $\mathbf{\sqrt{\overline{AB}^2 + \overline{BC}^2}}$

Which gives; AC = $\mathbf{\sqrt{5^2 + 12^2}}$ = 13

Length of EC = $\mathbf{\sqrt{\overline{CD}^2+ \overline{DE}^2}}$

Which gives; EC = $\mathbf{\sqrt{6^2 + 8^2}}$ = 10

Therefore, ΔACE is an isosceles triangle

Base of ΔACE = EC

Therefore;

Height of isosceles triangle ΔACE = $\mathbf{\sqrt{13^2 - 5^2}}$ = 12

Area of ΔACE = $\mathbf{\frac{1}{2}}$ × 10 × 12 = 60

Therefore;

• Area of the convex pentagon ABCDE = 30 + 24 + 60 = 104

Learn more about finding the area of geometric figures here:

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