Solving a Right Angled Triangle: Finding the Length of the Shorter Side
Right triangles, with their unique properties and applications, are significant in various fields such as mathematics, physics, and engineering. This tutorial will guide you through a step-by-step solution to find the length of the shorter side of a right-angled triangle, given specific conditions. Specifically, if the hypotenuse is 45 cm and the difference between the other two sides is 9 cm, we can use the Pythagorean theorem and quadratic equation to find the length of the shorter side. Let's dive into the problem and its solution.Given Conditions
We are given a right-angled triangle with the following conditions:
The length of the hypotenuse (c) is 45 cm. The difference between the two shorter sides (a - b) is 9 cm.Using these conditions, we need to find the length of the shorter side (a).
Step-by-Step Solution
Pythagorean Theorem: We start by using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.a2 b2 c2
Substituting the given hypotenuse length:
a2 b2 452 2025
Express a in terms of b: We know the difference between the two shorter sides:a - b 9
Substitute a in the Pythagorean equation: Now, we can express 'a' in terms of 'b' and substitute it in the Pythagorean equation.(b 9)2 b2 2025
Expand and simplify the equation: Expand the equation and combine like terms.b2 18b 81 b2 2025
2b2 18b - 1944 0
Divide the equation by 2: Simplify the quadratic equation by dividing all terms by 2.b2 9b - 972 0
Solve the quadratic equation: Use the quadratic formula to solve for 'b'.b (-B ± √(B2 - 4AC)) / 2A
Where A 1, B 9, and C -972.
Solving the Quadratic Formula
b (-9 ± √(92 - 4 × 1 × -972)) / (2 × 1)
b (-9 ± √(81 3888)) / 2
b (-9 ± √3969) / 2
b (-9 ± 63) / 2
b (54 / 2) 27 or b (-72 / 2) -36
Since lengths cannot be negative, we discard -36.
b 27 cm
Find the length of a: Using the difference equation, find the length of the longer side 'a'.a b - 9 27 - 9 18 cm
Conclusion
The length of the shorter side is 27 cm, and the length of the longer side is 36 cm. This solution verifies that (27, 36, 45) forms a Pythagorean triplet.