Triangles • Topic 4 of 6

Special Right Triangles (45-45-90, 30-60-90)

Because the GRE avoids trigonometry, it leans hard on two fixed-ratio right triangles, and memorising them is one of the highest-value moves in all of GRE prep. The 45-45-90 triangle (an isosceles right triangle, and half of a square along its diagonal) has sides in the ratio 1 : 1 : √2 — each leg times √2 gives the hypotenuse. The 30-60-90 triangle (half of an equilateral triangle) has sides in the ratio 1 : √3 : 2, where 1 is opposite the 30°, √3 is opposite the 60°, and 2 (the longest) is opposite the 90°. Anchor every 30-60-90 problem to the short leg (opposite 30°): double it for the hypotenuse, multiply it by √3 for the long leg. These show up whenever a square, an equilateral triangle, or a "nice" right triangle appears.

The two special right triangles: 45-45-90 and 30-60-90 with their side ratios45-45-9011√245°45°30-60-90√31230°60°Leg ratios 1:1:√2 and 1:√3:2

✅ Solved examples

1. A 45-45-90 triangle has legs of 5. Find the hypotenuse.
Hypotenuse = leg × √2 = 5√2.
2. A 45-45-90 triangle has a hypotenuse of 10. Find each leg.
Leg = hypotenuse / √2 = 10 / √2 = (10√2) / 2 = 5√2.
3. A 30-60-90 triangle has a short leg (opposite the 30° angle) of 4. Find the hypotenuse and the long leg.
Sides are in ratio 1 : √3 : 2 from the short leg. Hypotenuse = 2 × 4 = 8; long leg = 4√3.
4. A square has a diagonal of 8. Find the length of each side.
The diagonal splits the square into two 45-45-90 triangles with the diagonal as hypotenuse: side = 8 / √2 = (8√2) / 2 = 4√2.

✏️ Practice — try these, take hints as needed

1. A 45-45-90 triangle has legs of 7. Find the hypotenuse.
Hypotenuse = leg × √2.
Multiply 7 by √2.
Leave it in radical form.
7√2
2. A 30-60-90 triangle has a hypotenuse of 12. Find the short leg.
Short leg is opposite the 30°.
Hypotenuse = 2 × short leg.
Divide 12 by 2.
6
3. An equilateral triangle has a side of 6. Find its height.
The height makes a 30-60-90 triangle.
Short leg = 3, height is the long leg.
Long leg = short leg × √3.
3√3
4. A 45-45-90 triangle has a hypotenuse of 6√2. Find each leg.
Leg = hypotenuse / √2.
6√2 / √2.
The √2 cancels.
6
5. A 30-60-90 triangle has a long leg (opposite the 60° angle) of 5√3. Find the short leg.
Long leg = short leg × √3.
5√3 = short leg × √3.
Divide both sides by √3.
5

📝 Topic test — 8 questions

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