Mathematics — Standard (041) — CBSE Class 10 board question

Let the two sides be $a$ and $b$ cm.

Given: Perimeter = 60 cm, Hypotenuse = 25 cm

So, $a + b + 25 = 60$
$$a + b = 35 \quad \text{...(1)}$$

By Pythagoras theorem:
$$a^2 + b^2 = 25^2 = 625 \quad \text{...(2)}$$

From (1): $b = 35 - a$

Substituting in (2):
$$a^2 + (35 - a)^2 = 625$$
$$a^2 + 1225 - 70a + a^2 = 625$$
$$2a^2 - 70a + 600 = 0$$
$$a^2 - 35a + 300 = 0$$

Factorising:
$$a^2 - 20a - 15a + 300 = 0$$
$$(a - 20)(a - 15) = 0$$

So, $a = 20$ or $a = 15$

When $a = 20$, $b = 35 - 20 = 15$
When $a = 15$, $b = 35 - 15 = 20$

∴ The other two sides of the triangle are 20 cm and 15 cm.

Source: Chapter 4, Quadratic Equations (Factorisation method)

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• Examiners expect you to set up two equations from the given conditions (perimeter → linear; Pythagoras → quadratic).
• Substituting the linear equation into the quadratic and simplifying to standard form $ax^2 + bx + c = 0$ earns method marks.
• Factorising correctly and finding both roots, then interpreting them as the two sides, completes the answer.
• Always write a concluding statement — it is expected in CBSE board answers and carries the final mark.