Question

In ΔXYZ, m∠x: m∠Y: m∠Z = 1:2:3. If XY = 15, YZ = ......,select a proper option (a), (b), (c) or (d) from given options so that the statement becomes correct.
(a) 15√3/2
(b) 17
(c) 8
(d) 7.5

Answer 1

Given, In ΔXYZ, m∠x: m∠Y: m∠Z = 1:2:3.
Let proportionality constant is x
then, m∠X = x , m∠Y = 2x and m∠Z = 3x

now, m∠X + m∠Y + m∠Z = 180°
x + 2x + 3x = 180°
6x = 180°
x = 30°
then, m∠X = 30°, m∠Y = 60° and m∠Z = 90°

here m∠Z is a right angle so, ∆YZX is right angled triangle.

here you should understand one thing.
$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$
where A, B and C are the angle of triangle ABC also, a, b and c are the sides of corresponding angle.

here, for ∆XYZ,
$\frac{XY}{sinZ}=\frac{YZ}{sinX}=\frac{ZX}{sinY}$
here, XY = 15, we have to find YZ = ?
sinZ = sin90° = 1
sinX = sin30° = 1/2
so, 15/1 = YZ/(1/2)
=> 15 × 1/2 = YZ
=> YZ = 7.5
option (d) is correct.

Answer 2

Given :

In ∆XYZ ,

m<X : m<Y : m<Z = 1 : 2 : 3

XY = 15 ,

Let m<X = a ,

m<Y = 2a ,

m<Z = 3a ,

m<X + m<Y + m<Z = 180°

[ Angle sum property ]

=> a + 2a + 3a = 180°

=> 6a = 180°

=> a = 180/6 = 30

Therefore ,

m<X = a = 30°

m<Y = 2a = 2 × 30 = 60°

m<Z = 3a = 3 × 30 = 90°

∆XYZ is a right angle triangle.

Right angle at Z.

From figure attached above ,

In ∆XZY ,

sin X = YZ/XY

=> sin 30° = YZ/15

=> 1/2 = YZ/15

=> 15/2 = YZ

=> 7.5 = YZ

Therefore ,

YZ = 7.5

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