Three Side Lengths Can Tell You There Is a Right Angle

Yes. Test whether the squares of the two shorter sides add to the square of the longest. For sides 9, 12 and 15:

$$81+144=225=15^2$$

They match, so the triangle is right-angled — no protractor needed. This page covers everything in the ICSE Class 7 Mathematics chapter on the Pythagoras theorem: the statement and the hypotenuse, finding a missing side, the converse, and practical problems.

Identify the hypotenuse first, then either add the squares or subtract them.

Finding the hypotenuse — add. Legs of 6 cm and 8 cm:

$$c^2=6^2+8^2=36+64=100\Rightarrow c=\sqrt{100}=10\text{ cm}$$

Finding a leg — subtract. Hypotenuse 13 cm and one leg 5 cm:

$$b^2=13^2-5^2=169-25=144\Rightarrow b=\sqrt{144}=12\text{ cm}$$

Another. Hypotenuse 25 cm, one leg 7 cm:

$$b^2=625-49=576\Rightarrow b=24\text{ cm}$$

Whether to add or subtract is decided entirely by what is missing, so write down which side is the hypotenuse before touching the arithmetic. Looking for the hypotenuse means adding; looking for a leg means subtracting the smaller square from the larger.

A size check confirms the answer: the hypotenuse must come out larger than either leg. So 10 cm against legs of 6 and 8 is right, and a "hypotenuse" of 5 cm would be impossible.

That check also catches the commonest error, which is adding when you should subtract. Using $13^2+5^2=194$ above would give about 13.9 cm — a leg longer than the hypotenuse, which cannot happen.

The converse says that if the square of the longest side equals the sum of the squares of the other two, then the triangle is right-angled — and the right angle lies opposite the longest side.

To apply it: square all three sides, then check whether the two smaller squares add to the largest.

Test 9, 12, 15:

$$9^2+12^2=81+144=225\text{and}15^2=225$$

Equal, so the triangle is right-angled, with the right angle opposite the 15 cm side.

Test 5, 6, 8:

$$5^2+6^2=25+36=61\text{and}{8}^2=64$$

Not equal, so it is not right-angled.

Test 8, 15, 17:

$$64+225=289=17^2$$

Right-angled.

The comparison tells you more than yes or no. When the two smaller squares add to less than the largest, as in 5, 6, 8, the angle opposite the longest side is obtuse. When they add to more, that angle is acute.

The step that must come first is identifying the longest side, since only that one can be the hypotenuse. Testing $5^2+8^2$ against $6^2$ compares the wrong pair and will reject a triangle that is actually right-angled.

Sketch the situation, mark the right angle, and label which length is the hypotenuse.

Ladder against a wall. A ladder 13 m long rests with its foot 5 m from the wall. The wall is vertical and the ground horizontal, so the right angle is at the base and the ladder is the hypotenuse:

$${\text{height}}^2=13^2-5^2=169-25=144\Rightarrow \text{height}=12\text{ m}$$

Diagonal of a rectangle. A rectangle is 12 cm by 5 cm. The diagonal splits it into two right-angled triangles, so

$$d^2=12^2+5^2=144+25=169\Rightarrow d=13\text{ cm}$$

For a square of side 7 cm: $d^2=49+49=98$, so $d=\sqrt{98}$, about 9.9 cm.

Distance between two points. A man walks 9 m east, then 12 m north. His straight-line distance from the start is

$$\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15\text{ m}$$

He walked 21 m but ended only 15 m away, because east and north meet at a right angle.

The move that decides these questions is spotting which length is the hypotenuse, and it is always the one opposite the right angle — the ladder rather than the wall, the diagonal rather than a side, the direct route rather than either leg of the walk.