Question

an isosceles triangle has a base of 12 and two sides of 10. what other base measure can an isosceles triangle with equal sides of 10 have and still have the same area as the given triangle? (figure is not to scale.) an isosceles triangle with equal sides of 10 and the same area as the given triangle can have a base of . (simplify your answer.)

Explanation:

Step1: Find height of given triangle

First, use the Pythagorean theorem for the given isosceles triangle. The base is \(b = 12\), and the equal - sides are \(a = 10\). The height \(h\) of the given isosceles triangle can be found by considering half of the base (\(\frac{b}{2}=6\)). Using \(a^{2}=h^{2}+(\frac{b}{2})^{2}\), we get \(h=\sqrt{a^{2}-(\frac{b}{2})^{2}}=\sqrt{10^{2}-6^{2}}=\sqrt{100 - 36}=\sqrt{64}=8\). Then the area of the given triangle \(A=\frac{1}{2}\times b\times h=\frac{1}{2}\times12\times8 = 48\).

Step2: Let the base of the new isosceles triangle be \(x\) and find its height

Let the base of the new isosceles triangle with equal sides \(a = 10\) be \(x\). The height \(H\) of the new isosceles triangle using the Pythagorean theorem is \(H=\sqrt{10^{2}-(\frac{x}{2})^{2}}\).

Step3: Set up area equation

Since the area of the new triangle is the same as the given triangle, we have \(\frac{1}{2}\times x\times\sqrt{10^{2}-(\frac{x}{2})^{2}}=48\). Square both sides: \(\frac{1}{4}x^{2}(100-\frac{x^{2}}{4}) = 48^{2}\). Let \(t=\frac{x^{2}}{4}\), then the equation becomes \(t(100 - t)=48^{2}\), or \(100t-t^{2}=2304\), or \(t^{2}-100t + 2304=0\).

Step4: Solve the quadratic equation

For the quadratic equation \(t^{2}-100t + 2304=0\), where \(a = 1\), \(b=-100\), \(c = 2304\), the quadratic formula \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{100\pm\sqrt{(-100)^{2}-4\times1\times2304}}{2\times1}=\frac{100\pm\sqrt{10000 - 9216}}{2}=\frac{100\pm\sqrt{784}}{2}=\frac{100\pm28}{2}\). We get \(t_1=\frac{100 + 28}{2}=64\) and \(t_2=\frac{100 - 28}{2}=36\).

Step5: Find the value of \(x\)

If \(t = 64\), then \(\frac{x^{2}}{4}=64\), \(x^{2}=256\), \(x = 16\). If \(t = 36\), then \(\frac{x^{2}}{4}=36\), \(x^{2}=144\), \(x = 12\) (we already have the case of \(x = 12\) for the given triangle, so we take the other value).

Answer:

16