The height CH is drawn in a right-angled triangle ABC with right angle C. Find HB if AH = 3.2 and CH = 2.4.

Consider triangles ABC and ACН: both triangles are right-angled, angle A is common to both triangles. Therefore, triangle ABC is similar to triangle ACH (in two corners).

Consider triangles ABC and BCH: both triangles are right-angled, angle B is common to both triangles. Therefore, the ABC triangle is similar to the BCH triangle (in two corners).

Hence, the triangle ACH is similar to the triangle BCH.

Let us express the coefficient of similarity of triangles:

The similarity coefficient is: AH (leg of triangle ACH) / CH (leg of triangle BCH) = CH (second leg of triangle ACH) / BH (leg of triangle BCH).

AH / CH = CH / BH.

AH = 3.2, CH = 2.4.

Substitute the known data and calculate the BH value.

3.2 / 2.4 = 2.4 / BH.

BH = (2.4 * 2.4) / 3.2 = (2.4 * 24) / 32 = (2.4 * 3) / 4 = 0.6 * 3 = 1.8.

Answer: HB = 1.8.