Question

In ΔABC, ∠B = 90°. D is the mid-point of AB and DE || BC. If AB = 9 cm and AC = 15 cm, find the perimeter of DECB.

Answer 1

Step 1:

Use the Pythagorean theorem: AC² = AB² + BC²

Substitute the given values: 15² + 9² + BC²

Calculate BC: BC = `sqrt(15^2 - 9^2)`

= `sqrt(225 - 81)`

= `sqrt(144)`

= 12 cm

Step 2:

Since D is the mid-point of AB, DB = `1/2` × AB = `1/2` × 9 = 4.5 cm

Since DE || BC and D is the midpoint of AB, E must be the midpoint of AC by the converse of the Midpoint Theorem.

Therefore, DE = `1/2` × BC = `1/2` × 12 = 6 cm

Step 3:

Since E is the midpoint of AC,

EC = `1/2` × AC

= `1/2 xx 15`

= 7.5 cm

Step 4:

The perimeter of DECB = DE + EC + CB + BD

Substitute the calculated values: Perimeter = 6 + 7.5 + 12 + 4.5 = 30 cm.

The perimeter of DECB is 30 cm.