Question

In a right angled triangle the acute angles are in ratio 4:5.find the angles of the triangle in degrees and radian

Answer 1

Answer:

The angles of the Right Triangles are

∠A = 90° , 1.57 radian

∠B = 40° , 14.13 radian

∠C = 50° , 11.33 radian  

Step-by-step explanation:

Given as :

For a right angle triangle

The ratio of acute angle = 4 : 5

∵ In a right angle, one angle is 90°

Let in any Triangle ,

∠A = 90° and other angles are ∠B and ∠C

So, The ratio of ∠B and ∠C = ∠B : ∠C = 4 : 5

Let ∠B = 4 x

and ∠C = 5 x

And The measure of ∠A = 90°

Now, In any Triangle , sum of all three angles = 180°

So, ∠A + ∠B + ∠C = 180°                ......eq 1

So, Put the value of ∠A , ∠B and ∠C  in eq 1

i.e 90° + 4 x + 5 x = 180°

Or, 4 x + 5 x = 180° - 90°

Or, 9 x = 90°

∴  x =  $\frac{90^{\circ}}{9}$

So, x = 10°

So, The measure of angle  ∠B = 4 × 10° = 40°

And The measure of angle  ∠C = 5 x = 5 × 10° = 50°

So, All The three angles are

∠A = 90°

∵ 180° = π radian

∴ 90° = $\dfrac{\Pi }{180^{\circ}}$ × 90°

i.e  90° = $\dfrac{3.14}{2}$ = 1.57 radian

Again

∠B = 40°

∵ 180° = π radian

∴ 40° = $\dfrac{\Pi }{180^{\circ}}$ × 40°

i.e  40° = 14.13 radian

Similarly

∠C =  50°

∵ 180° = π radian

∴ 50° = $\dfrac{\Pi }{180^{\circ}}$ × 50°

i.e  50° = 11.33 radian

Hence, The angles of the Right Triangles are

∠A = 90° , 1.57 radian

∠B = 40° , 14.13 radian

∠C = 50° , 11.33 radian  Answer