Final answer:
The triangle with angle ratios of 3:2:1 is a right-angled triangle because the sum of its angles equals 180 degrees and the largest angle measures 90 degrees. Using the Pythagorean theorem with a hypotenuse of 25 cm, the length of the shortest side is calculated to be 12.5 cm.
Step-by-step explanation:
Understanding Right-Angled Triangles
To show that the triangle with angle ratios of 3:2:1 is a right-angled triangle, we must recognize that the sum of the angles in any triangle is always 180 degrees. Let's denote the smallest angle as X degrees. Therefore, the second angle will be 2X degrees, and the third angle will be 3X degrees. Adding these up gives us X + 2X + 3X = 180 degrees, or 6X = 180 degrees, which means X = 30 degrees. The largest angle will be 3X, which is 90 degrees, confirming it's a right-angled triangle.
To calculate the length of the shortest side in the triangle (let's call it 'a'), when the hypotenuse ('c') is 25 cm, we can use the Pythagorean theorem, which says a² + b² = c². We first need to find the ratio of the sides that correspond to the angles. Since 'c' is across from the right angle, it will be the largest side, meaning 'a' will be across from the smallest angle X (30 degrees). The side across from the 60 degrees angle will be √3 times the smallest side 'a', based on trigonometrical relationships. We only need to concern ourselves with 'a' for this question.
Given the Pythagorean theorem, if 'b' is √3 times 'a', then a² + (a√3)² = 25². Simplifying this gives us a² + 3a² = 625, which is 4a² = 625. Dividing both sides by 4 gives us a² = 156.25, and taking the square root of both sides yields a = 12.5 cm. Therefore, the length of the shortest side is 12.5 cm.