CAT 2025

Slot 3

QA

Question & SolutionQues & Sol

A triangle ABC is formed with AB = AC = 50 cm and BC = 80 cm. Then, the sum of the lengths, in cm, of all three altitudes of the triangle ABC is

1. Concept Used

• Topic: Geometry — Isosceles Triangle, Pythagorean Theorem, and Ratio of Altitudes

• Formula:

For a right-angled triangle,

$$ a^2+b^2=c^2 $$

The ratio of the altitudes of a triangle is inversely proportional to the ratio of the corresponding sides.

2. Calculation

Since $\triangle ABC$ is isosceles, the altitudes $CM$ and $BN$ are equal.

In $\triangle AOB$, using the Pythagorean Theorem:

$$ AO^2+OB^2=AB^2 $$

Therefore,

$$ AO^2=50^2-40^2 $$

$$ AO^2=2500-1600=900 $$

Hence,

$$ AO=30\text{ cm} $$

The ratio of the altitudes is inversely proportional to the ratio of the corresponding sides. Therefore,

$$ \frac{1}{50}:\frac{1}{50}:\frac{1}{80}=8:8:5 $$

Thus,

$$ CM=BN=8x $$

and

$$ AO=5x $$

Since $AO=30$,

$$ 5x=30 $$

$$ x=6 $$

Therefore, the sum of the altitudes is:

$$ 8x+8x+5x=21x $$

$$ =21\times6=126 $$

3. Solution

Answer = 126 cm ✅

The sum of the three altitudes is 126 cm.