Right Triangle Sides in AP, Area 24 | JEE Strategy Series

What happens when a right triangle's three side lengths increase by equal steps? In this worked book sample, the equal steps determine the shape. The area then determines the size.

For: JEE learners comfortable with expanding a square, the Pythagorean theorem and the area of a triangle. I am Dr. Biplab Pal, the author of the book linked below.

1. The complete book question

“A right triangle has sides in AP and area 24. Find its shortest side.”

AP means arithmetic progression: the same amount is added between neighbouring values. For example, 2, 5, 8 have equal steps of 3. A right triangle has one 90-degree angle, like a square corner.

Source: Develop Strategic Math Mind for JEE – Vol1, Complete Edition, book problem E090, printed page 107, one-based PDF page 107. This is the exact book wording. Its label “JEE Advanced 2017 Paper 1 Offline” is secondary book attribution; the official paper wording and official question number have not been independently verified. E090 is a book identifier. No named length or area unit is given.

2. What must the answer establish?

The triangle must satisfy three conditions: positive side lengths with equal steps, a right angle, and area 24. We need its shortest side, not just a ratio of the sides.

My reading of the skills tested is that the question combines an equal-step pattern with two geometry facts. The Pythagorean theorem relates the sides; the area formula fixes their scale. This is an inference from the question, not a claim about the examiner's private intent.

Name the increasing sides a − d, a, a + d, where d is positive. Equal sides cannot form a right triangle, so d cannot be zero. Also a − d must be positive.

Pause and think: Which side belongs alone on the right of the Pythagorean equation? Why should the area be used after the shape is found?

3. Work through the solution

The longest side is a + d. It lies opposite the right angle. Therefore:

(a − d)² + a² = (a + d)²

Expand both squares, cancel d², and collect the remaining terms:

a² − 2ad + d² + a² = a² + 2ad + d²
a² − 4ad = 0
a(a − 4d) = 0

Since a is a positive side length, a is not zero. Thus a = 4d, and the three sides are 3d, 4d, 5d.

The two shorter sides meet at the right angle. Their area is half their product:

½ × 3d × 4d = 24
6d² = 24
d² = 4, so d = 2.

We use the positive value because d is a positive length difference. The sides are therefore 6, 8 and 10.

Check every condition

• Equal steps: 8 − 6 = 2 and 10 − 8 = 2.
• Right angle: 6² + 8² = 36 + 64 = 100 = 10².
• Area: ½ × 6 × 8 = 24.

Answer: the shortest side is 6 length units. We do not invent centimetres when the book names no unit.

A likely mistake

It is tempting to put the middle side a alone in the Pythagorean equation. The side that belongs alone is the longest, a + d. And finding the ratio 3:4:5 is only part of the solution: the stated area must still be used.

A small change, fully solved

This is an added teaching question, separate from the quoted book problem: if the same equal-step right triangle had area 6, what would its shortest side be?

The shape is still 3d, 4d, 5d. Now 6d² = 6, so d = 1. The sides are 3, 4, 5, and the shortest side is 3 length units.

Try the book's approach before choosing it

This sample shows the approach: identify what each condition contributes, choose a useful representation, then check the answer. For another complete example, read the five-term geometric-progression product worked sample.

My Develop Strategic Math Mind for JEE – Vol1 is a paid English Kindle book. View the book and its Read sample option on Amazon India to judge the level and teaching style before buying. The print/PDF locator above is not a claim about a Kindle location number.