Question

XY is parallel to QR and PX/XQ = PY/YR = 1/2 find XY:QR

Answer 1

Answer:

$XY:QR=1:3$

Step-by-step explanation:

Given:

$\frac{PX}{XQ}=\frac{PY}{YR}=\frac{1}{2}$

Taking reciprocals

$\frac{YR}{PY}=2\\\\\frac{PY+YR}{PY}=1+2\\\\\frac{PR}{PY}=3$

In ΔPXY and ΔPQR,

∠P=∠P (common)

∠PXY=∠Q (corresponding angles)

∠PYX=∠R (corresponding angles)

ΔPXY and ΔPQR are equiangular

By AAA similarity,

ΔPXY and ΔPQR are similar

Their corresponding sides are proportional

$\frac{PX}{PQ}=\frac{XY}{QR}=\frac{PY}{PR}\\\\This\:\: implies\\\\\frac{XY}{QR}=\frac{PY}{PR}\\\\\frac{XY}{QR}=\frac{1}{3}\\\\XY:QR=1:3$

Answer 2

Answer:

1/3

Step-by-step explanation:

XY is parallel to QR and PX/XQ = PY/YR = 1/2 find XY:QR

XY ║ QR

∠PXY = ∠PQR

∠PYX = ∠PRQ

∠XPY = ∠QPR (common angle)

Δ PXY = Δ PQR

PQ/PX  = PR/PY  = QR/ XY

=> (PX + XQ)/PX  = (PY + YR)/PY  =  QR/ XY

=>  1 + XQ/PX  = 1 + YR/PY  = QR/XY

PX/XQ = PY/YR  = 1/2

=> XQ/PX = YR/PY = 2

=> 1 +2 = 1 +2 = QR/ XY

=> QR/XY = 3

=> XY/QR = 1/3

XY : QR = 1/3