CAT 2025 Slot 3 QA Question 16

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CAT 2025 Slot 3QAGeometry • TrianglesModerate

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

TITA Answer:

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Official Convenor Answer Key: 126

Answer and solution

Answer: 126

📌 Core Concept

The area of a triangle can be calculated using any side as the base and the corresponding altitude. For an isosceles triangle, the altitude from the apex to the base splits the base into two equal parts and can be found using the Pythagorean theorem.

🔢 Step-by-Step Solution

Identify the triangle type and given sides:

Triangle ABC is isosceles with AB = AC = 50 \text{ cm} and BC = 80 \text{ cm}.

Calculate the altitude from A to BC (h_a):

The base BC is 80 \text{ cm}, so half of BC is 40 \text{ cm}.

Using the Pythagorean theorem in triangle ABD (where D is the midpoint of BC):

AB^2 = AD^2 + BD^2 \implies 50^2 = h_a^2 + 40^2

2500 = h_a^2 + 1600 \implies h_a^2 = 900 \implies h_a = 30 \text{ \text{ cm}}

Calculate the area of triangle ABC:

Using base BC and height h_a:

\text{Area} = \frac{1}{2} \times BC \times h_a = \frac{1}{2} \times 80 \times 30 = 1200 \text{ \text{ cm}}^2

Calculate the altitude from B to AC (h_b):

Using base AB and height h_b:

\text{Area} = \frac{1}{2} \times AB \times h_b \implies 1200 = \frac{1}{2} \times 50 \times h_b

1200 = 25 \times h_b \implies h_b = \frac{1200}{25} = 48 \text{ \text{ cm}}

Calculate the altitude from C to AB (h_c):

Since ABC is isosceles, h_c = h_b = 48 \text{ cm}.

Sum of all three altitudes:

h_a + h_b + h_c = 30 + 48 + 48 = 126 \text{ \text{ cm}}

CAT 2025 Slot 3 QA Question 16

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