Is a triangle with sides 5, 12, and 13 a right triangle?
Understand the Problem
The question is asking whether a triangle with sides of lengths 5, 12, and 13 is a right triangle. This can be determined using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Answer
Yes, the triangle is a right triangle.
Answer for screen readers
The triangle with sides 5, 12, and 13 is a right triangle.
Steps to Solve
- Identify the sides of the triangle
We have sides of lengths 5, 12, and 13. The longest side, which is 13, will be treated as the hypotenuse.
- Apply the Pythagorean theorem
According to the Pythagorean theorem, for a triangle to be a right triangle, the following equation must hold: $$ c^2 = a^2 + b^2 $$ Where $c$ is the hypotenuse and $a$ and $b$ are the other two sides.
- Substitute the side lengths into the theorem
We substitute the values into the equation: $$ 13^2 = 5^2 + 12^2 $$
- Calculate the squares of the sides
Now calculate the squares: $$ 13^2 = 169 $$ $$ 5^2 = 25 $$ $$ 12^2 = 144 $$
- Add the squares of the two shorter sides
Now add the squares of the two shorter sides: $$ 25 + 144 = 169 $$
- Verify the equation
Now we check if the left side matches the right side: $$ 169 = 169 $$
Since both sides are equal, this confirms that the triangle with sides 5, 12, and 13 is indeed a right triangle.
The triangle with sides 5, 12, and 13 is a right triangle.
More Information
This conclusion is based on the Pythagorean theorem, which is a fundamental principle in geometry for right triangles. The specific triangle identified is a well-known example of a Pythagorean triple.
Tips
- Failing to identify the longest side as the hypotenuse.
- Miscalculating the squares of the side lengths.
- Forgetting to check whether $c^2$ equals $a^2 + b^2$ after substitution.
