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In the given Fig. PQR is a triangle, right angled at Q. If XY || QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.

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Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio. This is also known as Thales theorem.

Given:


\angle Q= 90^\circ , XY \ || \ QR, PQ = 6 \ \text{cm}, PY = 4 \ \text{cm} \ \text{and} \ PX : XQ = 1 : 2

Since,
XY \ || \ QR,


PX/XQ = PY/YR

[ By Thales theorem (BPT)]


(1)/(2) = PY/YR
[PX : XQ = 1 : 2]


(1)/(2) = 4 /(PR - PY)


[YR= PR - PY]


(1)/(2) = 4 /(PR - 4)


PR - 4 = 2 * 4


PR - 4 = 8


PR = 8 +4


PR = 12 \ \text{cm}

In right
\Delta PQR,


PR^2 = PQ^2 + QR^2

[ By Pythagoras theorem]


12^2 = 6^2 + QR^2
[\text{Given} : PQ= 6 \ \text{cm}]


144 = 36 + QR^2


144 - 36 + QR^2


108= QR^2


QR =√(108) =√(3*36) = 6√(3) \ \text{cm}

Hence, the lengths of PR and QR is 12 cm and
6√(3) cm.

User Mark Bonafe
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