Right triangles: hypotenuse and one side

The idea

If the hypotenuse and one side of a right triangle are proportional to the hypotenuse and one side of another right triangle, the two triangles are similar (RHS similarity criterion).

Two right triangles

Triangle ABC has a right angle at B, with hypotenuse AC = 5 and side AB = 3. Triangle DEF has a right angle at E, with hypotenuse DF = 10 and side DE = 6.

The hypotenuses are in the ratio 10/5 = 2 and the sides are in the ratio 6/3 = 2. The other side EF is not given.

Is Δ DEF similar to Δ ABC?

• Yes. The other sides must be in the ratio 2 too.
• No. We would also need to know EF.
• Only if EF is measured.

Build a right triangle

Triangle ABC is fixed, with the right angle at B. Build DEF with a right angle at E: choose the hypotenuse DF and the side DE with the dials. The other side EF follows.

Hypotenuse and one side

If in two right triangles the hypotenuse and one side of one are proportional to the hypotenuse and one side of the other, the triangles are similar (RHS criterion).

RHS stands for Right angle, Hypotenuse, Side. Both triangles already have a right angle. The proportional hypotenuse and side then fix the rest.

Let DF = k × AC and DE = k × AB. By the Pythagoras Theorem, EF² = DF² − DE² = k² × AC² − k² × AB² = k² × (AC² − AB²) = k² × BC². So EF = k × BC, and all three sides are in the same ratio k. By the SSS criterion, the triangles are similar.

Notes

If in two right triangles the hypotenuse and one side of one are proportional to the hypotenuse and one side of the other, the triangles are similar (RHS criterion).

Check yourself

In right triangle ABC (right angle at B), AC = 13 cm and AB = 5 cm. In right triangle DEF (right angle at E), DF = 26 cm and DE = 10 cm. Are the triangles similar?

In right triangle ABC (right angle at B), AC = 5 cm and AB = 3 cm. In right triangle DEF (right angle at E), DF = 10 cm and DE = 8 cm. Are the triangles similar?

Right triangle P has hypotenuse 5. Right triangle Q has hypotenuse 10. Are P and Q always similar?

Right triangle ABC (right angle at B) has AC = 10 cm, AB = 6 cm and BC = 8 cm. Right triangle DEF (right angle at E) has DF = 15 cm and DE = 9 cm. Find EF, in cm.

Answer: 12

DF/AC = DE/AB = 1.5, so Δ ABC ~ Δ DEF (RHS) and EF = 1.5 × BC = 1.5 × 8 = 12 cm.

• Yes, they are similar. — correct. Yes! 26/13 = 10/5 = 2. The hypotenuses and one side are in the same ratio, and both triangles have a right angle: RHS.
• No, we do not know EF.. RHS does not need the other side. It follows from the hypotenuse and one side.
• No, the triangles have different sizes.. Similar triangles can have different sizes.
• Yes, because both are right triangles and DF is twice AC.. The hypotenuses are in the ratio 2, but DE/AB = 8/3 is not 2. Both ratios must be equal.
• No, they are not similar. — correct. Yes! 10/5 = 2 but 8/3 is not 2. The hypotenuse and the side are not in the same ratio, so RHS does not hold.
• Yes, because DE is bigger than AB.. Being bigger is not enough. The two ratios must be equal.
• Yes. Both are right triangles and 10 is twice 5.. Knowing only the hypotenuses is not enough. Q could have sides 6 and 8, or sides 5 and about 8.7. These have different shapes.
• No. We also need one more side in the same ratio. — correct. Yes! A right triangle with hypotenuse 10 can have sides 6 and 8, or 5 and about 8.7. The shapes differ.
• Yes. All right triangles are similar.. No. A thin right triangle and a nearly isosceles one both have a right angle but different shapes.