Question

In a triangle ABC, three altitudes AD, BE and CF were drawn from the three vertices of the triangle. They intersect each other at point H, such that AH = 12 cm and BH = 9 cm. If the length of altitude AD = 15 cm, find the length of altitude BE.
1. 11.25 cm
2. 12 cm
3. 12.5 cm
4. 13 cm

Answer 1

Correct Answer - Option 4 : 13 cm

The intersection point H of the three altitudes is the orthocenter of the triangle

Now, the product of the length of the line segments of the altitudes divided by the orthocenter is equal for all three altitudes, i.e,

In triangle ABC,

⇒ AH × HD = BH × HE = CH × HF     ---- (1)

Given,

AH = 12 cm, BH = 9 cm, AD = 15 cm,

⇒ HD = AD – AH = 15 – 12 = 3 cm

Substituting in (1), we get,

⇒ 12 × 3 = 9 × HE

⇒ HE = 36/9 = 4 cm

∴ Length of altitude BE = BH + HE = 9 + 4 = 13 cm