Question

The bisectors of the acute angles of a right triangle meet at O. Find the angle at O between the two bisectors

Answer 1

Draw  a  triangle  as  shown in picture.

Let  ∠A = x , ∠C = y , ∠B = 90°

Now,

As  AO  and  CO are bisectors  of angle   x and  y, Therefore,

∠OAC = x/2

∠OCA = y/2

We know that, Sum of  interior angles  of  a triangle  is 180°

in ΔABC,

x + y + 90° = 180

x + y = 90°

in ΔAOC ,

x/2 + y/2 + ∠AOC = 180°

1/2(x+y) + ∠AOC  = 180°

90/2 + ∠AOC = 180°

∠AOC = 180 - 45

∠AOC = 135°

Answer 2

Given: ABC is the right angle triangle, the right angle will be formed at B.

AO and OC are the angle bisectors of

∠BAC and ∠BCA

To Find That:

∠AOC

Solution:

Since AO and OC are the angle bisectors of ∠BAC and ∠BCA

∠OAC = 1/2 ∠BAC - (1)

∠OCA = 1/2 ∠BCA - (2)

Now we will add equation 1 and 2

∠OAC + ∠OCA = 1/2 ∠BAC + 1/2∠BCA

= 1/2(∠BAC + ∠BCA)

= 1/2 (180-∠ABC)

(the sum of interior angles is 180 degrees)

∠OAC + ∠OCA = 1/2 [180-90]

= 1/2 * 90

= 45

- (3)

Now in the triangle AOC,

∠AOC = 180 - [∠OAC + ∠OCA]

= 180 - 45

= 135

So the angle at O between the two bisectors is 135 Degrees.