Question 4 The shortest leg of a triangle is 7 meters shorter than the other leg. The hypotenuse of this triangle is 17 meters. What are the lengths of the two legs of this triangle? The shortest leg is The other leg is Question Help: Video long. Submit Question

To find the lengths of the two legs of the triangle, we can use 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. Given: - The shortest leg is 7 meters shorter than the other leg. - The hypotenuse is 17 meters. Let's denote the length of the shortest leg as \( x \) and the length of the other leg as \( x + 7 \). Using the Pythagorean theorem: \[ x^2 + (x + 7)^2 = 17^2 \] Expanding the equation: \[ x^2 + x^2 + 14x + 49 = 289 \] Simplifying the equation: \[ 2x^2 + 14x + 49 = 289 \] Subtracting 289 from both sides: \[ 2x^2 + 14x - 240 = 0 \] Dividing the entire equation by 2 to simplify: \[ x^2 + 7x - 120 = 0 \] Now, we can solve this quadratic equation to find the values of \( x \), which will give us the lengths of the two legs of the triangle. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x^{2}+7x-120=0\) - step1: Factor the expression: \(\left(x-8\right)\left(x+15\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-8=0\\&x+15=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=8\\&x=-15\end{align}\) - step4: Rewrite: \(x_{1}=-15,x_{2}=8\) The solutions to the quadratic equation are \( x = -15 \) and \( x = 8 \). Since the length of a side cannot be negative, we discard the solution \( x = -15 \). Therefore, the length of the shortest leg is 8 meters, and the length of the other leg is \( 8 + 7 = 15 \) meters. So, the lengths of the two legs of the triangle are 8 meters and 15 meters.