# In a right-angled triangle ABC with a right angle C, the angle between the height CH and the bisector CD

In a right-angled triangle ABC with a right angle C, the angle between the height CH and the bisector CD is 15 °. Find the hypotenuse if AH = 6 and point D lies between points B and H.

Let’s define the acute angles of the ACН right-angled triangle.

CD = bisector of the right angle, therefore, the angle AC = 45, then the angle ACН = ACD – DCH = 45 – 15 = 30. Then the leg AH lies opposite the angle 30, and therefore is equal to half the length of the hypotenuse AC. AH = AC / 2.

AC = 2 * AH = 2 * 6 = 12 cm.

The angle СAН of the triangle ACН is equal to: СAН = 180 – 90 – 30 = 60, then the angle CAB = 60.

The angle ABC of the triangle ABC is equal to: ABC = 180 – ACB – CAB = 180 – 90 – 60 = 30.

The leg AC lies opposite the angle 30, therefore, its length is equal to half of the hypotenuse AB.

AC = AB / 2.

AB = 2 * AC = 2 * 12 = 24 cm.

Answer: The length of the hypotenuse AB is 24 cm.

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