In triangle ABC, the heights AA1 and CC1 intersect at point H. Find the height drawn to the side

In triangle ABC, the heights AA1 and CC1 intersect at point H. Find the height drawn to the side of AC if HA1 = 3cm. BA1 = 4cm. AH = 4cm

Consider a right-angled triangle ВНА1, in which, according to the Pythagorean theorem, we determine the length of the hypotenuse ВН.

BH ^ 2 = BA1 ^ 2 + HA1 ^ 2 = 4 ^ 2 + 3 ^ 2 = 16 + 9 = 25.

BH = 5 cm.

Let us prove the similarity of triangles ВНА1 and АНВ1.

Both triangles are rectangular. Angle АНВ1 = ВНА1 as vertical angles. Then right-angled triangles are similar in acute angle.

Then B1H / A1H = AH / BH.

B1H / 3 = 4/5.

B1H = 3 * 4/5 = 12/5 = 2.4 cm.

Then BB1 = BH + B1H = 5 + 2.4 cm = 7.4 cm.

Answer: The length of the height is 7.4 cm.