Question

In △PQR, X is the midpoint of QR. XY and XZ. Is the perpendicular drawn from X to PQ and PR respectively. If XY = XZ, prove that △ PQR is an isosceles triangle.​

Answer 1

$\bf\large\green{\underline{ given : }}$

• X is mid point of QR.
• XY and XZ are perpendicular to PQ and PR respectively.
• XY = XZ

$\bf\large\green{\underline{concept \: used : }}$

• RHS congruency . ( 2 sides are same and one of the angle is 90°)
• Sides opposite to equal angles is equal.

$\bf\large\green{\underline{ solution : }}$

In the attached diagram,

➜ In △YXQ and △ZXR ,

• Angle XYQ = Angle XZR = 90°
• YX = XZ ( given )
• XQ = XR ( As X is midpt. of QR )

∴△YXQ ≌△ZXR ( by RHS test )

So, Angle YQX = Angle ZRX

ㅤㅤㅤㅤㅤㅤ( in congruent triangles , all sides and angles are equal )

➜ In △PQR ,

• Angle PQR = Angle PRQ

And as we mentioned .....that sides opposite to equal angles are also equal.

1. Side opposite to Angle PQR is PR.
2. Side opposite to Angle PRQ is PQ.ㅤ

∴ PR = PQ ..........

• Isoscales triangle has two equal sides.

And in △PQR , PQ = PR..

∴ △PQR is isoscales triangle...