The height BD of a right-angled triangle ABC = 24cm and cuts off the segment DC = 18cm

The height BD of a right-angled triangle ABC = 24cm and cuts off the segment DC = 18cm from the hypotenuse AC.Find AB and cos A.

1. Calculate the length of the leg ВС of the right-angled triangle ВСD:

ВС = √ВD² + СD² = √24² + 18² = √576 +324 = √900 = 30 centimeters.

2. Calculate the length of the segment AD. For the calculation, we use the formula for calculating the height BD:

BD² = AD x CD.

AD = 576/18 = 32 centimeters.

3. AC = 32 + 18 = 50 centimeters.

4. Calculate the length of the leg AB:

AB = √AC² – BC² = √50² – 30² = √2500 – 900 = √1600 = 40 centimeters.

5. We calculate the cosine ∠А:

Cosine ∠A = AB / AC = 40/50 = 4/5.

Answer: cosine ∠A = 4/5.