The perimeter of a right triangle is 160, and the hypotenuse is 68. Find the area of the right triangle.

Given that the perimeter of a right triangle is 160 and the hypotenuse is 68, we can use the Pythagorean theorem to find the lengths of the other two sides of the triangle. Let's denote the lengths of the two sides as \( a \) and \( b \), and the hypotenuse as \( c \). The perimeter of the triangle is given by \( a + b + c = 160 \). Since it is a right triangle, we can use the Pythagorean theorem, which states that \( a^2 + b^2 = c^2 \). We are given that the hypotenuse \( c = 68 \). We can substitute this value into the Pythagorean theorem to find the lengths of the other two sides. Let's solve the system of equations to find the lengths of the sides of the right triangle. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}a+b+68=160\\a^{2}+b^{2}=68^{2}\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}a=92-b\\a^{2}+b^{2}=68^{2}\end{array}\right.\) - step2: Substitute the value of \(a:\) \(\left(92-b\right)^{2}+b^{2}=68^{2}\) - step3: Simplify: \(8464-184b+2b^{2}=68^{2}\) - step4: Move the expression to the left side: \(8464-184b+2b^{2}-68^{2}=0\) - step5: Calculate: \(3840-184b+2b^{2}=0\) - step6: Factor the expression: \(2\left(60-b\right)\left(32-b\right)=0\) - step7: Divide the terms: \(\left(60-b\right)\left(32-b\right)=0\) - step8: Separate into possible cases: \(\begin{align}&60-b=0\\&32-b=0\end{align}\) - step9: Solve the equation: \(\begin{align}&b=60\\&b=32\end{align}\) - step10: Calculate: \(b=60\cup b=32\) - step11: Rearrange the terms: \(\left\{ \begin{array}{l}a=92-b\\b=60\end{array}\right.\cup \left\{ \begin{array}{l}a=92-b\\b=32\end{array}\right.\) - step12: Calculate: \(\left\{ \begin{array}{l}a=32\\b=60\end{array}\right.\cup \left\{ \begin{array}{l}a=60\\b=32\end{array}\right.\) - step13: Check the solution: \(\left\{ \begin{array}{l}a=32\\b=60\end{array}\right.\cup \left\{ \begin{array}{l}a=60\\b=32\end{array}\right.\) - step14: Rewrite: \(\left(a,b\right) = \left(32,60\right)\cup \left(a,b\right) = \left(60,32\right)\) The solutions to the system of equations are \( (a, b) = (32, 60) \) or \( (a, b) = (60, 32) \). Now that we have the lengths of the sides of the right triangle, we can find the area of the triangle using the formula for the area of a right triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base and height of the triangle are the lengths of the two sides, which are 32 and 60, respectively. Let's calculate the area of the right triangle. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\times 32\times 60\) - step1: Multiply the terms: \(16\times 60\) - step2: Multiply the numbers: \(960\) The area of the right triangle is 960 square units.