Question

Explain two different methods for finding the length of the hypotenuse of a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle.

Short answer

The hypotenuse of a 45-45-90 triangle is \(x\sqrt{2}\), with \(x\) being the leg length.

Step-by-step solution

Understand the Properties of the 45-45-90 Triangle

A 45-45-90 triangle is an isosceles right triangle, meaning its two legs are of equal length. Let's denote the length of each leg as \(x\). The hypotenuse is opposite the right angle.

Method 1 - Use Pythagorean Theorem

For any right triangle, the Pythagorean theorem states that the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides. Thus:\[c^2 = x^2 + x^2 = 2x^2\]Taking the square root of both sides, the hypotenuse \(c\) is:\[c = \sqrt{2x^2} = x\sqrt{2}\]

Method 2 - Use Triangle Ratio

A 45-45-90 triangle has side ratios of 1:1:\(\sqrt{2}\). This means if each leg (\(x\)) is of length \(a\), then the hypotenuse is \(a\sqrt{2}\). Hence, for legs of length \(x\), the hypotenuse \(c\) is \(x\sqrt{2}\).

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written mathematically as \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.
To apply the Pythagorean Theorem in a 45-45-90 triangle, remember that both legs are of equal length. If we denote each leg as \(x\), then we use the formula: \[c^2 = x^2 + x^2 = 2x^2\] Taking the square root of both sides gives us the hypotenuse: \[c = \sqrt{2x^2} = x\sqrt{2}\]
This formula helps us find the length of the hypotenuse efficiently in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, showing the power of the Pythagorean Theorem in solving geometric problems.

isosceles right triangle

An isosceles right triangle is a special type of right triangle where two sides are of equal length. These two equal sides are the legs, and the angles opposite these sides are equal to \(45^{\circ}\) each. In a 45-45-90 triangle, knowing that it is isosceles helps streamline calculations significantly.

• The two angles other than the right angle are each \(45^{\circ}\).
• The two legs are identical in length, often denoted by \(x\).
• This configuration helps us utilize the inherent triangle properties and apply specific formulas.

Understanding these properties simplifies determining side lengths and helps apply methods like using side ratios or the Pythagorean theorem effectively, as you'll know exactly where they fit into the problem.

triangle side ratios

Triangle side ratios are extremely useful, especially in special triangles like the 45-45-90 triangle. Side ratios allow us to determine side lengths using proportions without needing complex calculations. For a 45-45-90 triangle, the side ratios are 1:1:\(\sqrt{2}\).

• This ratio tells us that the legs are the same length.
• The hypotenuse is \(\sqrt{2}\) times the length of either leg.
• If a leg length is \(x\), the hypotenuse will be \(x \sqrt{2}\).

By knowing these ratios, you can quickly determine missing sides if at least one side is known. This method complements the Pythagorean Theorem and offers a quick check or alternative approach to confirm your calculations, especially useful in exams or timed scenarios.