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.
Understanding Geometry: The 45-45-90 Triangle
The ratio works because the triangle has a strong symmetry. If you fold it along the line that splits the right angle in half, one leg matches the other leg exactly. This symmetry is useful because one known length determines every other length.
For example, if a floor tile is a square with side length 12 centimeters, its diagonal is 12 times the square root of two centimeters. If a diagonal is known instead, each side is the diagonal times the square root of two divided by two. This second form is often easier to use because it avoids leaving a square root in the denominator.
The Pythagorean theorem explains the pattern, but it is important to understand what the theorem is measuring. The two legs lie at right angles, so their squared lengths add to make the squared length of the longest side. With equal legs, the total is twice the square of one leg.
Taking the square root gives the extra factor of the square root of two. The hypotenuse is therefore about 1.414 times as long as either leg. That decimal is useful for estimating, though the exact square root form is better in geometry because it keeps answers precise.
Students often meet this triangle on coordinate grids. A segment that moves the same horizontal distance as vertical distance has a slope of positive one or negative one. A line with either of those slopes makes a 45 degree angle with a horizontal grid line.
For instance, a move of 5 units right and 5 units up creates a 45 degree direction. The straight line distance is 5 times the square root of two units.
This idea appears in distance formula problems, map grids, computer graphics, and paths across square game boards. It explains why a diagonal shortcut across a square is shorter than traveling along two sides, yet longer than one side.
Careful labeling prevents common mistakes. The hypotenuse is always opposite the right angle, and it is always the longest side. The equal sides are the legs, not the hypotenuse.
Do not use the special ratio merely because a drawing looks symmetrical. Confirm the angle measures or prove that the triangle comes from a square or another isosceles right figure. Keep units attached throughout a calculation, especially when working with area, scale drawings, or real measurements.
Finally, distinguish a length from its square. A diagonal length of 8 times the square root of two has a square equal to 128 square units, while the diagonal itself is measured in ordinary units.
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.