Question

The red is the answer I just need the steps that lead to the answer

Answer 1

The trapezoids LMNK and LMAB are simillar, which means that their sides are related by the same ratio. With this we can form the following equation:

`(2x-2)/(-15+6x)=(-15+6x)/(32)`

We need to solve for x.

`\begin{gathered} 32\cdot(2x-2)=(-15+6x)^2 \\ 64x-64=225-180x+36x^2 \\ 36x^2-180x-64x+225+64=0 \\ 36x^2-180x-64x+225+64=0 \\ 36x^2-244x+289=0 \end{gathered}`

Now we have to find the roots of the quadratic equation.

`\begin{gathered} x=\frac{-(-244)\pm\sqrt[]{(-244)^2-4(36)(289)}}{2\cdot36} \\ x=\frac{244\pm\sqrt[]{17920}}{64} \\ x=(244\pm133.87)/(64) \\ x=(244+133.87)/(64)=5.9 \\ x=(244-133.87)/(64)=1.72 \end{gathered}`

Therefore the midsegment is equal to:

`\begin{gathered} -15+6\cdot(5.9)=20.4_{} \\ or \\ -15+6\cdot(1.72)=-4.68 \end{gathered}`

Since the length can't be negative, the correct result is 20.4.