ACT Math: Use Special Right Triangles to Solve Problems Without a Calculator

A 45-45-90 triangle is isosceles with two legs of equal length. If each leg has length a, the hypotenuse has length a√2. Ratio: legs are 1:1, hypotenuse is √2. A 30-60-90 triangle has sides in the ratio 1:√3:2. If the side opposite 30° has length a, the side opposite 60° has length a√3, and the hypotenuse has length 2a. These two ratios unlock dozens of geometry problems on ACT Math without needing a calculator or the Pythagorean theorem.

Example 45-45-90: A right triangle has a 45° angle and a leg of length 5. The other leg is also 5 (isosceles), and the hypotenuse is 5√2. Example 30-60-90: A right triangle has a 30° angle, a 60° angle, and a 90° angle. If the side opposite 30° is 3, the side opposite 60° is 3√3, and the hypotenuse is 6.

For 45-45-90: Write the ratio as 1:1:√2. If you know one leg is 4, then the ratio becomes 4:4:4√2. For 30-60-90: Write the ratio as 1:√3:2. If you know the side opposite 30° is 2, multiply the entire ratio by 2: 2:2√3:4. This method—write the ratio, identify which value is known, scale the entire ratio—works for every special right triangle problem. No Pythagorean theorem needed; just scale the ratio.

Practice problem: A 45-45-90 triangle has a hypotenuse of 6. Find the legs. Ratio: 1:1:√2. Hypotenuse is 6, so √2 corresponds to 6. Solve: √2 times k=6, so k=6/√2=6√2/2=3√2. Each leg is 3√2. Verify: (3√2)^2+(3√2)^2=18+18=36, and √36=6. Correct.

(1) A square has a diagonal of 10. Find the side length. (A square is two 45-45-90 triangles.) Diagonal is hypotenuse=10. Ratio 1:1:√2 scales to ?:?:10. Side=10/√2=5√2. (2) An equilateral triangle has a side of 8. Find the height. (Height creates two 30-60-90 triangles.) Height is opposite 60°. In 30-60-90 with side=8 opposite 30°, height=8√3. (3) A right triangle has a 30° angle and hypotenuse 12. Find the side opposite 30°. Ratio 1:√3:2 scales to ?:?:12. Side opposite 30°=12/2=6. (4) A 45-45-90 triangle has legs of x. Find the hypotenuse in terms of x. Hypotenuse=x√2. (5) A 30-60-90 triangle has a hypotenuse of 5. Find the side opposite 60°. Ratio 1:√3:2 scales to 5/2:(5√3)/2:5. Side opposite 60°=(5√3)/2. Work through these five using the ratio-scaling method and you'll own special right triangles.

Drill: Create your own five problems where you're given one side and must find another. Use the ratio-scaling method for each. Check answers using the Pythagorean theorem to verify.

On every ACT Math section, 2-4 questions involve special right triangles, often hidden in geometry or coordinate problems. Students who know the ratios solve these instantly without a calculator. Students who don't know the ratios either use the Pythagorean theorem (slower) or guess (wrong). Each special right triangle question you solve correctly is a point that took you 30 seconds instead of 2 minutes.

Memorize the two ratios today. Write them on your scratch paper at the start of the test. Spend one week drilling the five-problem set and verifying answers with the Pythagorean theorem. By test day, you'll recognize special right triangles instantly and solve them faster than any peer using a calculator.