Question

The length of each side of an equilateral Δ ABC is 8cm. P and Q are the mid points of the sides AB

and AC respectively. Find the value of the length PQ and ∠ APQ. Give me the method ​

Answer 1

Answer:

PQ = 4cm and ∠APQ = 60°

Step-by-step explanation:

given an equilateral triangel, ABC of length = 8cm

=> ∠A = ∠B = ∠C = 60° --------- (1)

also, given that P and Q are midpoints of AB and AC resply,

=> PQ || BC --------- (2)

(by the property, the line joining the midpoints of any two sides of a triangle is parallel to the third side.)

if considered AB is the transversal of the parallel lines, PQ and BC,

=> ∠B = ∠P --------- (3) ( corresponding angles are equal )

similarly, taken AC as the transversal of the parallel lines, PQ and BC,

=> ∠C = ∠Q --------- (4) ( corresponding angles are equal )

so, by (1),(3) and (4),

=> ∠P = ∠Q = 60°

ie., ∠APQ = 60° & ∠AQP = 60° ------------ (5)

the same can be considered as,

in ΔAPQ , ∠A = ∠P = ∠Q = 60°

=> ΔAPQ is an equilateral triangle --------(6)

given P is midpoint of AB(8cm), so, AP = AB/2 = 4cm

ie., AP = AQ = PQ = 4cm , by (6)

so, the soln is PQ = 4cm and ∠APQ = 60°

Answer 2

Given: The length of each side of an equilateral Δ ABC is 8 cm. P and Q are the mid points of the sides AB and AC respectively.

To find: The value of the length PQ and ∠ APQ.

Solution:

According to the mid-point theorem, "The line segment joining the mid-points of any two sides of a triangle is parallel to the third side, and is equal to half of it."

The third side of the triangle is BC and is 8 cm in length according to the question. PQ is the line segment joining the mid-points of the two sides AB and AC of a triangle.

Hence,

$PQ = \frac{1}{2} BC$

      $= \frac{1}{2} * 8$

      $= 4 cm$

Since ABC is an equilateral triangle, all its angles are equal to 60°. Angles ∠ABC and ∠APQ are corresponding angles and thus, they are equal. So, ∠APQ is equal to 60°.

Therefore, the value of the length PQ is 4 cm and ∠APQ is 60°.