Questions: Special Right Triangles: 45-45-90

Question 1

The legs of a 45-45-90 triangle are each 7 cm. What is the hypotenuse?

• A. 14 cm
• B. 7√3 cm
• C. 7√2 cm
• D. 7/√2 cm

Answer: (C)

In a 45-45-90 triangle, hypotenuse = leg × √2. So 7 × √2 = 7√2 cm. Option A (14 cm) is the most common error — students multiply the leg by 2 instead of √2, confusing this with doubling. Option B (7√3) comes from the 30-60-90 triangle, where the long leg = short leg × √3.

Question 2

The hypotenuse of a 45-45-90 triangle is 10 cm. What is the length of each leg?

• A. 5 cm
• B. 10√2 cm
• C. 5√2 cm
• D. 10/√3 cm

Answer: (C)

When the hypotenuse is given, each leg = hypotenuse ÷ √2. Rationalizing: 10/√2 = 10√2/2 = 5√2 cm. Option A (5 cm) is wrong — it halves the hypotenuse instead of dividing by √2. Option B multiplies instead of divides. The key move is dividing by √2 and rationalizing, not halving.

Question 3

In a 45-45-90 triangle, the hypotenuse is twice the length of each leg.

• T. True
• F. False

Answer: False

The hypotenuse is √2 times the leg, not twice. Since √2 ≈ 1.414, the hypotenuse is about 41% longer than a leg — significantly less than double. This is one of the most persistent misconceptions. The factor of 2 applies to the *square* of the sides (1² + 1² = 2), not to the sides themselves.

Question 4

A 45-45-90 triangle can be produced by cutting a square along its diagonal.

• T. True
• F. False

Answer: True

Cutting a unit square diagonally produces two congruent triangles, each with two legs of length 1 (the square's sides) and angles of 45-45-90 (since the square's corners are 90° and the diagonal bisects each symmetrically). Applying the Pythagorean theorem gives the hypotenuse as √(1² + 1²) = √2. This is the cleanest derivation of the 1:1:√2 ratio.

Question 5

Where does the √2 in the 45-45-90 side ratio come from? Explain the derivation.

Model answer: In an isosceles right triangle with legs of length 1, the Pythagorean theorem gives c² = 1² + 1² = 2, so c = √2. The √2 arises because both legs are equal, meaning you are squaring the same number twice and adding — which is equivalent to multiplying that square by 2, then taking the square root.

The derivation is pure Pythagorean theorem on equal legs. The key insight is that the hypotenuse is not simply the sum of the legs (which would give 2) but rather the square root of the sum of their squares (√2). This is why the ratio is 1:1:√2 and not 1:1:2 — a distinction that matters in every applied problem.