Question

a line and two triangles are shown on the coordinate plane.which statements are true? select all that apply.the slope of (overline{jl}) is equal to the slope of (overline{kl}).the slope of (overline{jl}) is equal to the slope of (overline{ml}).the slope of (overline{kl}) is equal to the slope of the line.the slope of the line is equal to (\frac{ml}{jm}).the slope of the line is equal to (\frac{jm}{ml}).the slope of the line is equal to (\frac{jl}{ml}).

Explanation:

Step1: Recall the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For a line, the slope is constant. Let's assume two - point form for different line segments.
Let \(K=(0,2)\), \(L=(4,8)\). Then the slope of \(\overline{KL}\) is \(m_{KL}=\frac{8 - 2}{4-0}=\frac{6}{4}=\frac{3}{2}\).
Let \(J=(4,8)\), \(M=(- 2,8)\), \(N=(0,8)\).
The slope of \(\overline{JL}\): Using \(J=(4,8)\) and \(L=(4,8)\) (wait, no, assume \(J\) is not the same as \(L\) in terms of non - vertical/horizontal. Let's re - define.
Let's use the concept that for a line, the slope between any two points on the line is the same.
The slope of a line \(y=mx + b\) (where \(m\) is the slope) can also be represented as \(m=\frac{\text{rise}}{\text{run}}\).
For the line in the coordinate - plane, if we consider two right - triangles formed with the line. The slope of a line segment \(\overline{AB}\) with \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
The slope of \(\overline{KL}\): \(K=(0,2)\), \(L=(4,8)\), \(m_{KL}=\frac{8 - 2}{4-0}=\frac{6}{4}=\frac{3}{2}\).
The slope of the line: Using two points on the line (say \((0,2)\) and \((4,8)\)) is \(m=\frac{8 - 2}{4-0}=\frac{3}{2}\).
The slope of \(\overline{ML}\): \(M=(-2,8)\), \(L=(4,8)\) (no, wrong. Let's use the right - triangle. If \(JM\) is the horizontal change and \(ML\) is the vertical change in a right - triangle.
The slope of a line \(m=\frac{\text{vertical change}}{\text{horizontal change}}\).
The slope of \(\overline{KL}\) (where \(K=(0,2)\) and \(L=(4,8)\)): \(m=\frac{y_L - y_K}{x_L - x_K}=\frac{8 - 2}{4-0}=\frac{6}{4}=\frac{3}{2}\).
The slope of the line (using the same formula for any two points on the line) is the same as the slope of \(\overline{KL}\).
If we consider the right - triangle with vertical side \(ML\) and horizontal side \(JM\), the slope of the line \(m = \frac{ML}{JM}\) (since \(m=\frac{\text{rise}}{\text{run}}\)).

Step2: Check each statement

• Statement 1: The slope of \(\overline{JL}\) is equal to the slope of \(\overline{KL}\)

Assume \(J\) is a point on the line. Since the line has a constant slope, the slope of \(\overline{JL}\) (where \(J\) and \(L\) are on the line) is equal to the slope of \(\overline{KL}\) (because the slope of a line is constant. \(m=\frac{y_2 - y_1}{x_2 - x_1}\), and for any two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line, the ratio \(\frac{y_2 - y_1}{x_2 - x_1}\) is the same).

• Statement 2: The slope of \(\overline{KL}\) is equal to the slope of the line

Since \(\overline{KL}\) is a line segment on the line, by the definition of the slope of a line (the slope between any two points on the line is the same), the slope of \(\overline{KL}\) is equal to the slope of the line.

• Statement 3: The slope of the line is equal to \(\frac{ML}{JM}\)

By the formula \(m=\frac{\text{rise}}{\text{run}}\), where \(ML\) is the vertical change (\(\text{rise}\)) and \(JM\) is the horizontal change (\(\text{run}\)) in the right - triangle formed with the line.

Answer:

The slope of \(\overline{JL}\) is equal to the slope of \(\overline{KL}\); The slope of \(\overline{KL}\) is equal to the slope of the line; The slope of the line is equal to \(\frac{ML}{JM}\)