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 its 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: - Area of the isosceles triangle = \(60 \, \text{cm}^2\) - Length of each equal side = 13 cm Let's denote: - The base of the isosceles triangle as \(b\) - The height of the isosceles triangle as \(h\) The formula for the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] For an isosceles triangle, the height divides the base into two equal parts. Therefore, the height is the perpendicular bisector of the base. Given that the area of the triangle is \(60 \, \text{cm}^2\) and the length of each equal side is 13 cm, we can use the Pythagorean theorem to find the height of the triangle. Let's denote the height as \(h\). Then, using the Pythagorean theorem: \[ h^2 = 13^2 - \left(\frac{b}{2}\right)^2 \] Now, we can substitute the values into the area formula and solve for the base \(b\). 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 \( b = 10 \, \text{cm} \) or \( b = 24 \, \text{cm} \). Therefore, the correct answer is (A) 24 cm or 10 cm.