In a right-angled triangle ABC, the angle is A = 90 degrees, AB = 20cm. Height АD = 12cm. Find AC and cos C.

A right-angled triangle is a triangle with one of its angles equal to 90 °.

The height drawn from the vertex of the right angle of a right-angled triangle divides it into two triangles like this one.

Thus, ΔABD is similar to ΔADC.

In similar triangles, the three sides of one triangle are proportional to the three sides of the other. Means:

AC / AB = AD / BD = DC / AD.

In order to find these proportions, you need to find the side length BD. To do this, we use the Pythagorean theorem:

AB ^ 2 = AD ^ 2 + BD ^ 2;

BD ^ 2 = AB ^ 2 – AD ^ 2;

BD ^ 2 = 20 ^ 2 -12 ^ 2 = 400 – 144 = 256;

ВD = √256 = 16 cm.

Now, with the help of proportion, we find the side of the AC:

AC / 20 = 12/16;

AC = 20 12/16 = 15 cm.

Let’s find the side of DC:

DC / 12 = 12/16;

DC = 12 ∙ 12/16 = 144/16 = 9 cm.

To calculate cos C, we use the cosine theorem, according to which the cosine is the ratio of the adjacent leg to the hypotenuse:

cos C = DC / AC;

cos C = 9/15 = 0.6.

Answer: AC length = 15 cm; cos C = 0.6.