Study guides › Level 7 · Geometry

Special right triangles

Level 7 · Geometry · for questions like “In a 30°-60°-90° triangle the shortest side is 5. How long is the longest side?”

💡 The idea

Two right triangles come up so often that their side ratios are worth knowing:

• 45°-45°-90°: the sides are in the ratio 1 : 1 : √2.
• 30°-60°-90°: the sides are in the ratio 1 : √3 : 2. The shortest side is across from 30°, and the longest (hypotenuse) is across from 90°.

🪜 The steps

1. Find which side you know, and which number in the ratio it matches.
2. Scale the ratio by the same amount.

✏️ Worked examples

1. 45°-45°-90°, each leg 7: longest side = 7√2.

2. 30°-60°-90°, shortest side 5: longest side = 10, and the middle side = 5√3.

⚠️ Watch out

• The hypotenuse is always across from the 90° angle.
• In a 30°-60°-90° triangle, √3 goes with the middle side, not the longest.

🎯 Try these

1. 45°-45°-90°, legs of 4. Longest side?
2. 30°-60°-90°, shortest side 9. Longest side?
3. 30°-60°-90°, shortest side 6. Middle side?

Answers: 1) 4√2 2) 18 3) 6√3

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.