MathematicsIn a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and AB= 30 cm, the perimeter of the quadrilateral ARPQ is ____ cm.

In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and AB= 30 cm, the perimeter of the quadrilateral ARPQ is ____ cm.

Solution:

In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and AB= 30 cm, the perimeter of the quadrilateral ARPQ is 51 cm.
Given a triangle in which P, Q and R are the mid-points of sides BC, CA and AB respectively and AC = 21 cm, BC = 29 cm, AB= 30 cm.

Using mid-point theorem,

R P = 12 A C R P | | A Q, R P = A Q

Hence, ARPQ is a parallelogram.
The measure of side is,

A R = A B 2 ⇒ A R = 302 ⇒ A R = 15 cm

Opposite sides are equal,
AR = QP
⇒ QP = 15 cm
Also,

R P = A C 2 ⇒ R P = 212 ⇒ R P = 10.5 cm

Also, opposite sides are equal
RP = AQ
⇒ AQ = 10.5 cm
The perimeter is,
P = sum of all sides

⇒ P = A R + Q P + R P + A Q ⇒ P = 15 + 15 + 10.5 + 10.5 ⇒ P = 30 + 21 ⇒ P = 51 cm

Hence, the perimeter is 51 cm.