If the lengths of the sides of a triangle are in the ratio 3:4:5, what type of t

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Question: If the lengths of the sides of a triangle are in the ratio 3:4:5, what type of triangle is it?

Options:

Correct Answer: Right

Solution:

A triangle with sides in the ratio 3:4:5 satisfies the Pythagorean theorem, thus it is a right triangle.

If the lengths of the sides of a triangle are in the ratio 3:4:5, what type of triangle is it?

If the lengths of the sides of a triangle are in the ratio 3:4:5, what type of triangle is it?

Step 1: Understand that the sides of the triangle are in the ratio 3:4:5. This means if we let the sides be 3x, 4x, and 5x for some positive number x, we can find the actual lengths of the sides.
Step 2: Recall the Pythagorean theorem, which states that in a right triangle, the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.
Step 3: Identify the longest side in the ratio 3:4:5, which is 5x. The other two sides are 3x and 4x.
Step 4: Apply the Pythagorean theorem: (5x)² = (3x)² + (4x)².
Step 5: Calculate: (5x)² = 25x², (3x)² = 9x², and (4x)² = 16x².
Step 6: Add the squares of the two shorter sides: 9x² + 16x² = 25x².
Step 7: Since 25x² = 25x² is true, the triangle satisfies the Pythagorean theorem.
Step 8: Conclude that since it satisfies the Pythagorean theorem, the triangle is a right triangle.

Triangle Ratios – Understanding the significance of side length ratios in determining the type of triangle.
Pythagorean Theorem – Applying the Pythagorean theorem to identify right triangles based on side lengths.