In triangle ABC, the heights AA1 and CC1 meet at point H. Find the height to side AC if HA1

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

Let’s build the height of BB1 to the AC side. The heights of the triangle, or their extensions, intersect at one point.

In a right-angled triangle ВНА1, according to the Pythagorean theorem, we determine the length of the hypotenuse ВН. BH ^ 2 = A1H ^ 2 + A1B ^ 2 = 9 + 16 = 25.

BH = 5 cm.

The right-angled triangles BHA1 and AHB1 are similar in acute angle, since the angle BHA1 = AHB1 as vertical angles.

Then, from the similarity of triangles: BH / HA1 = AH / HB1.

HB1 = HA1 * AH / BH = 3 * 4/5 = 2.4 cm.

BB1 = BH + HB1 = 5 + 2.4 = 7.4 cm.

Answer: The length of the height is 7.4 cm.