Question

The length of the side of a triangle are 8,15, and 17 . Show that the triangle is a right triangle.

Short answer

The triangle with side lengths 8, 15, and 17 is a right triangle because it satisfies the Pythagorean theorem, as shown by the calculations: \(a^2 = 8^2 = 64\), \(b^2 = 15^2 = 225\), and \(c^2 = 17^2 = 289\). Since \(64 + 225 = 289\), the theorem holds, proving that the triangle is a right triangle.

Step-by-step solution

Identify the largest side

The side lengths of the triangle are 8, 15, and 17. The largest side is 17.

Apply Pythagorean theorem

Let's denote the side lengths as follows: a = 8, b = 15, and c = 17 (where c is the hypotenuse). According to the Pythagorean theorem, a right triangle must satisfy the equation a² + b² = c².

Check if the equation holds

Calculate the squares of the sides and check if the sum of the squares of the shorter sides is equal to the square of the possible hypotenuse: \(a^2 = 8^2 = 64\) \(b^2 = 15^2 = 225\) \(c^2 = 17^2 = 289\) Now, let's check if a² + b² equals c²: \(64 + 225 = 289\) Since the sum of the squares of the shorter sides (64 + 225) is equal to the square of the hypotenuse (289), the triangle with sides of lengths 8, 15, and 17 is indeed a right triangle.

Key concepts

Right Triangle

A right triangle is a special kind of triangle where one angle measures exactly 90 degrees. This right angle forms the heart of many geometric concepts and calculations. Knowing one angle is a right angle, we can use specific theorems, like the Pythagorean Theorem, to explore and solve problems related to such triangles.
Right triangles are often used in various fields, from architecture to navigation, because they simplify complex calculations. The two sides forming the right angle are perpendicular to each other, known as the "legs," whereas the side opposite the right angle is the longest side of the triangle.

Hypotenuse

The hypotenuse is the longest side in a right triangle, opposite the right angle. It plays a vital role in the Pythagorean Theorem, which is used to determine whether a triangle is a right triangle and to find missing side lengths.
In our case, with triangle sides of 8, 15, and 17, the largest number, 17, is recognized as the hypotenuse. This understanding is crucial because the Pythagorean Theorem only applies to the relationship between the legs of a triangle and the hypotenuse. Therefore, recognizing which side is the hypotenuse allows us to verify right triangles accurately.

Triangle Sides

The sides of a triangle are its defining features. In a right triangle, the relationship between these sides adheres to specific geometric rules. The two shorter sides, or "legs," support the hypotenuse and are essential in calculations.
When evaluating a set of side lengths to determine if they form a right triangle, as with 8, 15, and 17, it's important to identify them correctly:

• Assign the largest number as the hypotenuse, if using the Pythagorean Theorem.
• Validate that the sum of the squares of the two shorter sides equals the square of the hypotenuse.

This method guarantees the triangle's classification as a right triangle.

Proof of Right Triangle

Proving a triangle is a right triangle is straightforward with the Pythagorean Theorem. This theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides equals the square of the hypotenuse.
For example, with sides 8, 15, and 17:

• Calculate the squares: 8² = 64, 15² = 225, and 17² = 289.
• Add 64 and 225 to get 289, which matches 17².

Since both are equal, the triangle is confirmed as a right triangle. This proof is valuable for validating geometric properties and solving real-world problems efficiently.