Discover how to use the Pythagorean Theorem to solve a right-angled triangle problem. Specifically, determine the minimum length of the shorter leg to ensure that the hypotenuse is at least 13 cm. This article covers the step-by-step solution, including setting up the equation, solving the inequality, and validating the results.

Introduction

A right-angled triangle problem can be approached using the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is a fundamental concept in trigonometry and is extensively used in real-life applications, such as construction, engineering, and navigation.

Problem Statement

The problem at hand is to determine the minimum length of the shorter leg of a right-angled triangle such that the hypotenuse is at least 13 cm. The longer leg is 7 cm longer than the shorter leg.

Solution

Setting Up the Equation

Let's denote the length of the shorter leg as ( x ) cm. Consequently, the length of the longer leg is ( x 7 ) cm. According to the Pythagorean theorem, we have:

[ x^2 (x 7)^2 c^2 ]

Ensuring the Hypotenuse is at Least 13 cm

To ensure the hypotenuse ( c ) is at least 13 cm, we set up the inequality:

[ x^2 (x 7)^2 geq 13^2 ]

Calculations and Inequality Setup

First, calculate ( 13^2 ):

[ 13^2 169 ]

Substitute this into the inequality:

[ x^2 (x 7)^2 geq 169 ]

Expanding and Simplifying

Expand ( (x 7)^2 ):

[ (x 7)^2 x^2 14x 49 ]

Substitute this back into the inequality:

[ x^2 x^2 14x 49 geq 169 ]

Combine like terms:

[ 2x^2 14x 49 geq 169 ]

Subtract 169 from both sides:

[ 2x^2 14x 49 - 169 geq 0 ]

This simplifies to:

[ 2x^2 14x - 120 geq 0 ]

Divide the entire inequality by 2:

[ x^2 7x - 60 geq 0 ]

Solving the Quadratic Inequality

To solve the quadratic inequality ( x^2 7x - 60 geq 0 ), we first factor the quadratic.

[ x^2 7x - 60 (x 12)(x - 5) ]

The critical points are found by setting each factor to zero:

[ x 12 0 quad Rightarrow quad x -12 ]

[ x - 5 0 quad Rightarrow quad x 5 ]

Test the intervals around the critical points ( x -12 ) and ( x 5 ):

For ( x [ (-13 12)(-13 - 5) (1)(-18) -18 geq 0 , text{(True)} ] For ( -12 [ (0 12)(0 - 5) 12(-5) -60 geq 0 , text{(False)} ] For ( x > 5 ), choose ( x 6 ): [ (6 12)(6 - 5) 18(1) 18 geq 0 , text{(True)} ]

The solution to the inequality ( (x 12)(x - 5) geq 0 ) is:

[ x in (-infty, -12] cup [5, infty) ]

Since ( x ) represents a length, it must be non-negative:

[ x geq 5 ]

Conclusion

Thus, the shortest leg must be at least 5 cm to ensure that the hypotenuse is at least 13 cm. This solution validates the use of the Pythagorean theorem in real-world applications and demonstrates the importance of proper inequality solving techniques.

By ensuring the shortest leg is at least 5 cm, the triangle's hypotenuse will meet the required length of at least 13 cm.