In a right-angled triangle ABC, the height CD is lowered from the vertex of the right angle C

In a right-angled triangle ABC, the height CD is lowered from the vertex of the right angle C. Projection side AC = 10 cm, AD = 4 cm, calculate the hypotenuse AB.

The height CD divided our triangle into two right-angled triangles ADC and DBC.
Take triangle ADC. According to the Pythagorean theorem:
AD ^ 2 + DC ^ 2 = AC ^ 2
DC ^ 2 = AC ^ 2 – AD ^ 2 = 10 * 10 – 4 * 4 = 100 – 16 = 84.

Let’s denote DB as y and BC as x. Then AB = AD + y.

Take the DBC triangle. By the Pythagorean theorem, we compose the equation y ^ 2 + CD ^ 2 = x ^ 2
Take triangle ABC. Similarly, we have: AC ^ 2 + x ^ 2 = (AD + y) ^ 2.
So, we have a system of equations:
y ^ 2 + 84 = x ^ 2
100 + x ^ 2 = (4 + y) ^ 2
In the second equation, instead of x ^ 2, substitute the value y ^ 2 + 84 and expand the parentheses, we get:

100 + y ^ 2 + 84 = 16 + 8 * y + y ^ 2
100 + 84 – 16 = 8 * y + y ^ 2 – y ^ 2
8y = 168
y = 21.
AB = 21 + 4 = 25 cm.