Question

In trainglePQR and triangleXYZ, angleP = angleY, angleR = angleX and PQ = YZ. Then, the two triangles are​

Answer 1

Answer:

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.