in ∆XYZ and ∆PQR ,it is given that ∆PQR is congruent to ∆XYZ, angle Y + angle Z=90° and xy:xz = 3:4 . Find the ratio of sides in ∆PQR. I'll mark u as branliest
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Answer:
Given that, ∆pqr ~ ∆xyz, so we have
Since the two triangles are similar, therefore, they will have their corresponding angles congruent and the corresponding sides in proportion.
pq/qr = pr/xz = qr/yz …… (i)
and,
∠p = ∠x ….. (ii)
also,
xy:yz = 3:4 …… (iii) [given in the question]
Consider ∆xyz,
∠y + ∠z = 90° [given in the question]
∴ ∠x + ∠y + ∠z = 180° [∵ sum of angles of a triangle is 180°]
∠= 180° - 90° = 90°
∴ ∠p = 90° ….. (iv) [from eq. (ii)]
Now, consider ∆pqr,
∠p + ∠q + ∠r = 180°
⇒∠q + ∠r = 180° - 90° = 90°
or, ∠q = ∠r = 45°
∴ ∆pqr is an isosceles triangle [since two angles of an isosceles triangle are equal]
∴ pq = pr …… (v) [∵ two sides opposite to equal angles of a triangle are also equal]
From the equations (i), (iii) & (v), we can deduce that
pq : qr : pr = 3 : 4 : 3
Hence, the ratio of the sides pq, qr & pr in the triangle pqr is 3 : 4 : 3.
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