The area of an isosceles triangle is \( 60 \mathrm{~cm}^{2} \) and the length of each one of its equal sides is 13 cm . Find ts base. \( \begin{array}{llll}\text { A) } 24 \mathrm{~cm} \text { or } 10 \mathrm{~cm} & \text { (B) } 20 \mathrm{~cm} \text { or } 8 \mathrm{~cm} & \text { (C) } 18 \mathrm{~cm} \text { or } 6 \mathrm{~cm} & \text { (D) } 16 \mathrm{~cm} \text { or } 4 \mathrm{~cm}\end{array} \)

Given: - The area of the isosceles triangle is \(60 \, \text{cm}^2\). - The length of each equal side of the isosceles triangle is 13 cm. To find the base of the isosceles triangle, we can use the formula for the area of a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Since it is an isosceles triangle, the height can be found using the Pythagorean theorem: \[ \text{height} = \sqrt{(\text{side})^2 - (\text{base}/2)^2} \] Let's denote the base of the triangle as \(b\). We can set up the equation using the given area and solve for the base. Solve the equation by following steps: - step0: Solve for \(b\): \(60=\frac{1}{2}b\sqrt{13^{2}-\left(\frac{b}{2}\right)^{2}}\) - step1: Find the domain: \(60=\frac{1}{2}b\sqrt{13^{2}-\left(\frac{b}{2}\right)^{2}},-26\leq b\leq 26\) - step2: Simplify: \(60=\frac{b\sqrt{676-b^{2}}}{4}\) - step3: Swap the sides: \(\frac{b\sqrt{676-b^{2}}}{4}=60\) - step4: Cross multiply: \(b\sqrt{676-b^{2}}=4\times 60\) - step5: Simplify the equation: \(b\sqrt{676-b^{2}}=240\) - step6: Raise both sides to the \(2\)-th power\(:\) \(\left(b\sqrt{676-b^{2}}\right)^{2}=240^{2}\) - step7: Evaluate the power: \(676b^{2}-b^{4}=57600\) - step8: Move the expression to the left side: \(676b^{2}-b^{4}-57600=0\) - step9: Factor the expression: \(\left(-b+24\right)\left(b-10\right)\left(b+10\right)\left(b+24\right)=0\) - step10: Separate into possible cases: \(\begin{align}&-b+24=0\\&b-10=0\\&b+10=0\\&b+24=0\end{align}\) - step11: Solve the equation: \(\begin{align}&b=24\\&b=10\\&b=-10\\&b=-24\end{align}\) - step12: Check if the solution is in the defined range: \(\begin{align}&b=24\\&b=10\\&b=-10\\&b=-24\end{align},-26\leq b\leq 26\) - step13: Find the intersection: \(\begin{align}&b=24\\&b=10\\&b=-10\\&b=-24\end{align}\) - step14: Check the solution: \(\begin{align}&b=24\\&b=10\end{align}\) - step15: Rewrite: \(b_{1}=10,b_{2}=24\) The solutions for the base of the isosceles triangle are \(b_1 = 10 \, \text{cm}\) and \(b_2 = 24 \, \text{cm}\). Therefore, the base of the isosceles triangle is either 10 cm or 24 cm. The correct answer is: A) 24 cm or 10 cm