In a triangle ABC, AB = 13 cm, BC = 20 cm, BD is the height of the triangle, BD = 12 cm. Calculate the lengths of the projections

In a triangle ABC, AB = 13 cm, BC = 20 cm, BD is the height of the triangle, BD = 12 cm. Calculate the lengths of the projections of the sides AB, BC onto the line AC and the length of the side AC.

If the height BD is drawn in the triangle ABC, then the triangles ABD and BCD are rectangular.

The projection of the AB side onto AC is the segment AD.

The projection of the BC side onto the AC is the DC segment.

Then, by the Pythagorean theorem, we find AD and DC.

AD ^ 2 = AB ^ 2 – BD ^ 2;

AD ^ 2 = 13 ^ 2 – 12 ^ 2;

AD ^ 2 = 169 – 144;

AD ^ 2 = 25;

AD = 5.

DC ^ 2 = BC ^ 2 – BD ^ 2;

DC ^ 2 = 20 ^ 2 – 12 ^ 2;

DC ^ 2 = 400 – 144;

DC ^ 2 = 256;

DC = 16.

AC = AD + DC.

AC = 5 + 16 = 21.

Answer: 5cm, 16cm, 21cm.