P, Q and R are the mid-points of the sides of an equilateral triangle XYZ. Then PQR is

an equilateral triangle

Reason

Given,

XYZ is an equilateral triangle.

⇒ XY = YZ = ZX

P, Q and R are the mid-points of the sides of an equilateral triangle XYZ.

By mid-point theorem,

The line joining mid-point of any two sides of a triangle is parallel and is equal to half of third side.

Since, P and Q are mid-points of sides XY and XZ,

∴ PQ || YZ and PQ = $\dfrac{1}{2}$ YZ

(As we know, XY = YZ = ZX)

⇒ PQ = $\dfrac{1}{2} YZ = \dfrac{1}{2}XY = \dfrac{1}{2}$ XZ ……………….(1)

And, Q and R are mid-points of sides XZ and YZ,

∴ QR || XY and QR = $\dfrac{1}{2}$ XY

(As we know, XY = YZ = ZX)

⇒ QR = $\dfrac{1}{2} XY = \dfrac{1}{2}YZ = \dfrac{1}{2} XZ$ ……………….(2)

Similarly, P and R are mid-points of sides XY and YZ,

∴ PR || XZ and PR = $\dfrac{1}{2}$ XZ

(As we know, XY = YZ = ZX)

⇒ PR = $\dfrac{1}{2}XZ = \dfrac{1}{2} XY = \dfrac{1}{2}$ YZ ……………….(3)

From equation (1), (2) and (3), we can clearly say that

⇒ PQ = QR = PR = $\dfrac{1}{2}XZ = \dfrac{1}{2} XY = \dfrac{1}{2}$ YZ

⇒ PQ = QR = PR

Hence, PQR is also an equilateral triangle.