Question

In right triangle ABC, the altitude CH to the hypotenuse AB intersects the angle bisector AL at point D. Find the sides of ABC if AD = 8 cm and DH = 4 cm?

Answer 1

In right triangle ABC, the sides are AB = 12 cm, BC = 4 cm, and AC = 8 cm.

Step-by-step explanation:

In triangle ABC, the altitude CH to the hypotenuse AB intersects the angle bisector AL at point D. Given that AD = 8 cm and DH = 4 cm, we can find the sides of triangle ABC.

First, we can use the properties of similar triangles to determine the lengths of AD and DH in relation to the sides of the triangle. Since triangle CDH is similar to triangle CAB, we have:

• AD/AC = DH/BC
• 8/AC = 4/BC
• AC/BC = 2

Using this ratio, we can express the lengths of AC and BC in terms of a common variable. Let's assume that AC = 2x and BC = x. Now, we can use the Pythagorean theorem and the angle bisector theorem to find the lengths of AB, BC, and AC:

• AB = AC + BC = 2x + x = 3x
• BD/DC = AB/AC
• BC/DC = (AB - AC)/AC

Now, we can substitute the values we know to find the lengths of the sides:

• BD/DC = 3x/2x = 3/2
• BC/DC = (3x - 2x)/2x = 1/2

Since we know that BD = DH + BH = 4 + x and DC = CH + HD = 4 + 4 = 8, we can set up the following equation:

• (4 + x)/8 = 1/2

Simplifying, we find x = 4. Substituting this value back into our expressions for the side lengths, we have:

• AB = 3x = 3(4) = 12 cm
• BC = x = 4 cm
• AC = 2x = 2(4) = 8 cm