Question

which proportion can be used to show that the slope of jl is equal to the slope of mp?
triangle mnp and triangle jkl are similar right triangles.

Explanation:

Step1: Recall the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For line \(JL\), if we consider two points \(J(x_J,y_J)\) and \(L(x_L,y_L)\), and for line \(MN\), two points \(M(x_M,y_M)\) and \(N(x_N,y_N)\). Since \(\triangle MNP\sim\triangle JKL\), the ratios of their corresponding sides are equal.

Step2: Apply the slope - similarity relationship

The slope of a line segment between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{\text{rise}}{\text{run}}\). For \(\triangle JKL\), the slope of \(JL\) is \(m_{JL}=\frac{JK}{KL}\), and for \(\triangle MNP\), the slope of \(MN\) is \(m_{MN}=\frac{MP}{NP}\). Because of the similarity of \(\triangle MNP\) and \(\triangle JKL\), \(\frac{JK}{KL}=\frac{MP}{NP}\) (corresponding sides of similar triangles are proportional).

Answer:

\(\frac{JK}{KL}=\frac{MP}{NP}\)