Question

In a right angle Δ XYZ right angle at Y, YZ = 12, and  XZ = 13. The area of the triangles is 39 sq. units. Find the measurement of ∠X and ∠Z.
1. 60° , 30° 
2. 50° , 40° 
3. 45° , 45° 
4. 40° , 50° 

Answer 1

Correct Answer - Option 1 : 60° , 30° 

Given∶

In right angle ΔXYZ, YZ = 12, XZ = 13 and Area of Δ = 39 sq. units.

Formula Used∶ 

Sinθ = (Area × 2)/(a × b), where a and b are the measurements of 2 sides of a triangle.

Calculation∶

Use the formula,

Area = 1/2 × YZ × XZ × Sin(Z)

⇒ Sin Z = (2 × Area)/(YZ × XZ)

⇒ Sin Z = (2 × 39)/(12 × 13) = 0.5

Taking the inverse,

Z = Sin^-1(0.5) = 30° 

That is , ∠Z = 30 ° 

∠Y = 90      [given]

By using the triangle Angle sum Theorem, The measure of ∠X.

∠X = 180° - (90° + 30° ) = 180° - 120° = 60° 

∴ The measurement of ∠ X is 60°  and of ∠ Z is 30° .