Question

ABC is a right-angled triangle with hypotenuse AC = 13 cm and side AB = 5 cm. Perpendiculars are drawn from mid-point M of AC to AB and BC. What is the perimeter of the resulting quadrilateral?

Answer 1

Step 1:

Use the Pythagorean theorem:

BC² = AC² – AB²

BC² = 13² – 5²

= 169 – 25

= 144

BC = `sqrt(144)` = 12 cm

Step 2:

Let P be the point on AB and Q on BC.

MP is parallel to BC and MQ is parallel to AB by the midpoint theorem.

`MP = 1/2 xx BC` 

= `1/2 xx 12`

= 6 cm

`MQ = 1/2 xx AB`

= `1/2 xx 5`

= 2.5 cm

Since MP ⊥ AB and MQ ⊥ BC and ∠B = 90°, BPMQ is a rectangle.

BP = MQ = 2.5 cm

BQ = MP = 6 cm

Step 3:

Perimeter = BP + PM + MQ + QB

Perimeter = 2.5 + 6 + 2.5 + 6 = 17 cm