In a right-angled triangle, the leg is 12, and the hypotenuse is 13. Find the bisector of the triangle

In a right-angled triangle, the leg is 12, and the hypotenuse is 13. Find the bisector of the triangle drawn from the vertex of the smaller angle.

AB = 12, BC = 13.

By the Pythagorean theorem in the triangle ABC we find AC.

The sum of the squares of the legs is equal to the square of the hypotenuse:
a ^ 2 + b ^ 2 = c ^ 2

12 ^ 2 + b ^ 2 = 13 ^ 2

b ^ 2 = 13 ^ 2 – 12 ^ 2

b ^ 2 = 169 – 144

b ^ 2 = 25

b = 5

We conclude that the angle ABC is a smaller angle.

cosABС = 12/13 = 0.92

according to the table of cosines, we find that the angle ABC = 22 °.

The bisector of a triangle is a segment that connects the vertex with the opposite side and divides the corresponding angle in half.

Hence half is 11 °.

We get a new triangle AВD.

Let’s find the length of the bisector:

ВD = AB * √ ((2 * BC) / (AB + BC))

ВD = 12 * √ ((2 * 13) / (12 + 13))

ВD = 12 * √ (26/25)

ВD = 12 * 1.019

ВD ≈ 12.23

Answer: ВD ≈ 12.23