What is the length of the bisector of the smallest angle of a 3-4-5 right triangle?

Question

 I would like to know the length of the bisector of the smallest angle of a 3-4-5 right triangle please thanks

Answer 1

Outline of steps:

1. The smallest angle will be opposite the shortest side, let's call it angle A. Use one of the trig functions, say cosine, to determine the size of angle A. (Sketching the triangle may be helpful.)
2. Then calculate half of angle A; let's call this E.
3. The right angle is formed by the sides of length 3 and 4. So, we have another right triangle formed by the sides of lengths 3 and 4 and the bisector of angle A. Let's let x represent the unknown length of the angle bisector, which will be the hypotenuse of this smaller triangle. (Add the angle bisector and angle E to the sketch of your triangle from step 1.)
4. Now write an equation using the cosine of E (and the adjacent side 4 and hypotenuse x).
5. Solving the equation for x will answer your question. (Hint: the answer is between 4 and 4.3.)

 

Answer 2

Draw and label of diagram:

C is the right angle

B is the smallest angle

A is the middle sized angle

Draw BD the bisector of ∠B.

By the Angle Bisector Theorem

CD / AD = BC / BA

CD / AD = 4 / 5

CA = 3

CD = 4/3

Δ BCD is a right triangle. Use Pythagoras to determine BD.