In triangle GHI, if the lengths of sides GH, HI, and GI are in the ratio 3:4:5,

In triangle GHI, if the lengths of sides GH, HI, and GI are in the ratio 3:4:5, what type of triangle is it? (2019)

In triangle GHI, if the lengths of sides GH, HI, and GI are in the ratio 3:4:5, what type of triangle is it? (2019)

Step 1: Identify the sides of triangle GHI based on the given ratio 3:4:5. Let's say GH = 3x, HI = 4x, and GI = 5x for some positive number x.
Step 2: Recall the Pythagorean theorem, which states that in a right-angled 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.
Step 3: In our triangle, the longest side is GI, which is 5x. The other two sides are GH (3x) and HI (4x).
Step 4: Apply the Pythagorean theorem: (5x)² = (3x)² + (4x)².
Step 5: Calculate: 25x² = 9x² + 16x².
Step 6: Simplify the right side: 25x² = 25x².
Step 7: Since both sides of the equation are equal, this confirms that triangle GHI satisfies the Pythagorean theorem.
Step 8: Therefore, triangle GHI is a right-angled triangle.

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