Question

the line shown has a slope of $\frac{1}{2}$ between point j and point l and between point r and point t
which of the following best describes triangle jkl and triangle rst?
○ a. triangle jkl is similar to triangle rst.
○ b. the length of $overline{jl}$ is equal to the length of $overline{rt}$
○ c. triangle jkl is congruent to triangle rst
○ d. the length of $overline{jk}$ is equal to the length of $overline{rs}$

Explanation:

Step1: Recall the slope formula and similar triangles

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For two right - angled triangles (formed by the line and the grid), if the slopes are equal, the ratios of their corresponding sides are equal.
Since the slope between \(J\) and \(L\) and between \(R\) and \(T\) is \(\frac{1}{2}\), for \(\triangle JKL\) and \(\triangle RST\) (both right - angled triangles), \(\frac{JK}{KL}=\frac{RS}{ST}=\frac{1}{2}\) (by the definition of slope, where slope \(m=\frac{\text{rise}}{\text{run}}\)). Also, the right angles are equal (\(\angle JKL=\angle RST = 90^{\circ}\)).
By the Side - Angle - Side (SAS) similarity criterion (if the ratio of two pairs of corresponding sides of two triangles is equal and the included angles are equal, the triangles are similar), \(\triangle JKL\sim\triangle RST\).

Step2: Analyze other options

• For option B: Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(JL=\sqrt{(x_J - x_L)^2+(y_J - y_L)^2}\) and \(RT=\sqrt{(x_R - x_T)^2+(y_R - y_T)^2}\). From the grid, the lengths of \(JL\) and \(RT\) are not equal.
• For option C: Congruent triangles require \(JK = RS\), \(KL=ST\) and \(JL = RT\) (by SSS criterion). But from the grid (by counting the units for the sides of the right - angled triangles), \(JK

eq RS\), \(KL
eq ST\) and \(JL
eq RT\).

• For option D: By counting the units on the grid (horizontal units for the sides of the right - angled triangles), \(JK

eq RS\).

Answer:

A. Triangle \(JKL\) is similar to triangle \(RST\)