If a triangle has sides of lengths 3, 4, and 5, is it a right triangle?

If a triangle has sides of lengths 3, 4, and 5, is it a right triangle?

• A. Yes
• B. No
• C. It depends on the angles
• D. Only if the longest side is opposite the right angle

Answer: (A)

To determine if a triangle with sides of lengths 3, 4, and 5 is a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. In this case, the longest side is 5. We square all three sides: - The square of 3 is \(3^2 = 9\). - The square of 4 is \(4^2 = 16\). - The square of 5 is \(5^2 = 25\). Now, we check if the sum of the squares of the two shorter sides equals the square of the longest side: \(9 + 16 = 25\). Since both sides of the equation are equal, this confirms that the triangle with sides 3, 4, and 5 indeed satisfies the condition for being a right triangle. Thus, the answer is affirmed as accurate: yes, it is a right triangle. Understanding the Pythagorean theorem is key for this type of question, and recognizing the definition of a right triangle aligns perfectly with the conditions outlined by this theorem.

To determine if a triangle with sides of lengths 3, 4, and 5 is a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side.

In this case, the longest side is 5. We square all three sides:

• The square of 3 is (3^2 = 9).

• The square of 4 is (4^2 = 16).

• The square of 5 is (5^2 = 25).

Now, we check if the sum of the squares of the two shorter sides equals the square of the longest side:

(9 + 16 = 25).

Since both sides of the equation are equal, this confirms that the triangle with sides 3, 4, and 5 indeed satisfies the condition for being a right triangle. Thus, the answer is affirmed as accurate: yes, it is a right triangle.

Understanding the Pythagorean theorem is key for this type of question, and recognizing the definition of a right triangle aligns perfectly with the conditions outlined by this theorem.