Question

In triangle pqr and triangle xyz it is given that triangle pqr is similar to triangle xyz angle y+anglez equal to 90 degrees and xy:yz=3:4 then find the ratio of sides in triangle pqr

Answer 1

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/xy = qr/yz = pr/xz …… (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°]

⇒ ∠x = 180° - 90° = 90°

∴ ∠p = 90° ….. (iv) [from eq. (ii)]

Also,

xy:yz = 3:4 and ∠x is a right angle, therefore we can say that

⇒ yz is hypotenuse

By using the Pythagoras theorem, we get

xz = $\sqrt{4^2 - 3^2 }= \sqrt{7}$

⇒ xy : yz : xz = 3 : 4 : √7

Similarly,

since Δ PQR is a similar triangle to Δ XYZ

From (iv), we have p is a right angle

⇒ qr is the hypotenuse

⇒ pr : qr : pq = 3 : 4 : √7

Hence, the ratio of the sides pr, qr & pq in the triangle pqr is 3 : 4 : √7.  

Answer 2

Answer:

Given,

Δxyz ≅ Δpqr

PQ=4cm ; QR=5cm ; XY=6cm

⇒\frac{PQ}{XY}

XY

PQ

= \frac{QR}{YZ}

YZ

QR

[corresponding sides of similar triangles are in the same ratio]

⇒ \frac{4}{5} = \frac{6}{YZ}

5

4

=

YZ

6

⇒YZ = \frac{30}{4}YZ=

4

30

⇒YZ = 7.5 cm