Triangles • Topic 5 of 5

Special Right Triangles

Two right triangles have fixed side ratios worth memorising. In a 45-45-90 triangle the two legs are equal and the hypotenuse is a leg times √2, so a leg of 5 gives a hypotenuse of 5√2. In a 30-60-90 triangle the sides are in the ratio 1 : √3 : 2, where the shortest side faces the 30° angle, the hypotenuse (longest) is twice the shortest, and the side facing 60° is the short side times √3. These let you find all sides from one. They appear in squares (diagonals split into 45-45-90) and equilateral triangles (split into 30-60-90), and frequently on the SAT.

A 45-45-90 triangle and a 30-60-90 triangle with their side ratiosSpecial right triangles45°45°sss√245°-45°-90°60°30°xx√32x30°-60°-90°

✅ Solved examples

1. A 45-45-90 triangle has legs of 6. Find the hypotenuse.
Leg × √2 = 6√2.
2. A 30-60-90 triangle has shortest side 4. Find the hypotenuse.
Twice the shortest: 2 × 4 = 8.
3. A square has side 5. Find its diagonal.
A 45-45-90 split: 5√2.
4. A 30-60-90 triangle has shortest side 7. Find the hypotenuse.
2 × 7 = 14.

✏️ Practice — try these, take hints as needed

1. A 45-45-90 triangle has legs of 9. Find the hypotenuse.
Leg × √2.
—
—
9√2.
2. A 30-60-90 triangle has shortest side 5. Find the hypotenuse.
Hypotenuse = 2 × shortest.
2 × 5.
—
10.
3. A square has side 8. Find its diagonal.
Diagonal = side × √2.
—
—
8√2.
4. A 30-60-90 triangle has shortest side 6. Find the hypotenuse.
2 × 6.
—
—
12.
5. A 45-45-90 triangle has legs of 3. Find the hypotenuse.
3 × √2.
—
—
3√2.

📝 Topic test — 8 questions

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