A 45-45-90 triangle is a special right triangle with two equal acute angles and one right angle. Because the two acute angles are both 45 degrees, the legs opposite them are congruent. This triangle appears whenever a square is cut along its diagonal, making it one of the most useful patterns in geometry.
Knowing its side ratio helps students solve many problems without using a calculator every time.
The key relationship is that the two legs are equal and the hypotenuse is the length of a leg multiplied by sqrt(2). This comes from the Pythagorean theorem: if each leg is x, then the hypotenuse satisfies c^2 = x^2 + x^2 = 2x^2, so c = x sqrt(2). The ratio of the side lengths is 1 : 1 : sqrt(2).
This ratio is used in square diagonals, coordinate geometry, trigonometry, construction, and physics problems involving equal perpendicular components.
Key Facts
- A 45-45-90 triangle has angle measures 45 degrees, 45 degrees, and 90 degrees.
- The two legs are congruent because they are opposite equal 45 degree angles.
- The side length ratio is leg : leg : hypotenuse = 1 : 1 : sqrt(2).
- If each leg is x, then the hypotenuse is x sqrt(2).
- If the hypotenuse is h, then each leg is h / sqrt(2) or h sqrt(2) / 2.
- A square with side length s has diagonal length d = s sqrt(2).
Vocabulary
- 45-45-90 triangle
- A right triangle with two 45 degree angles and one 90 degree angle.
- Leg
- One of the two sides that form the right angle in a right triangle.
- Hypotenuse
- The side opposite the right angle and the longest side of a right triangle.
- Congruent
- Congruent figures or segments have exactly the same size and shape or the same length.
- Square diagonal
- A segment connecting opposite corners of a square, which splits the square into two congruent 45-45-90 triangles.
Common Mistakes to Avoid
- Using 1 : 2 : sqrt(3) for a 45-45-90 triangle is wrong because that ratio belongs to a 30-60-90 triangle.
- Multiplying the hypotenuse by sqrt(2) to find a leg is wrong because the hypotenuse is already longer than each leg; divide by sqrt(2) instead.
- Labeling the hypotenuse as one of the equal sides is wrong because the hypotenuse must be opposite the 90 degree angle and is the longest side.
- Forgetting that both legs are equal is wrong because the equal 45 degree angles guarantee the legs have the same length.
Practice Questions
- 1 A 45-45-90 triangle has legs of length 8 cm. Find the hypotenuse in exact form.
- 2 The diagonal of a square is 12 sqrt(2) inches. Find the side length of the square.
- 3 A right triangle has side lengths 7, 7, and 7 sqrt(2). Explain how you know it is a 45-45-90 triangle.