Mathematics
In the adjoining figure, ABCD is a square. P, Q and R are points on the sides AB, BC and CD respectively such that AP = BQ = CR and ∠PQR = 90°. Prove that
(a) △PBQ ≅ △QCR
(b) PQ = QR
(c) ∠PRQ = 45°
Triangles
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Answer
(a) Given, ABCD is a square.
∴ AB = BC.
Given, AP = BQ.
∴ AB - AP = BC - BQ …….(i)
In △PBQ and △QCR,
BQ = CR (Given)
From (i) we get,
PB = QC
∠PBQ = ∠QCR (Both are equal to 90°, as each angle in square = 90°.)
Hence, proved △PBQ ≅ △QCR by SAS axiom.
(b) We know that corresponding parts of congruent triangles are equal.
∴ PQ = QR.
Hence, proved that PQ = QR.
(c) Considering △PQR.
We know PQ = QR and ∠Q = 90°.
Hence, △PQR is an isosceles triangle with ∠P = ∠R = x.
Sum of angles of triangle = 180°.
⇒ ∠P + ∠Q + ∠R = 180°
⇒ x + 90° + x = 180°
⇒ 2x = 90°
⇒ x = 45°.
Hence, proved that ∠PRQ = 45°.
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