Question

1. in the graph below, triangles jhi and mlk are right triangles, and points i, k, j and m all lie on a straight line. which statement is true about the graph? a. the absolute value of the slope of the line is ml / jh. b. the measure of angle l is less than the measure of angle h. c. the absolute value of the slope of the line is equal to ih / hj. d. the absolute value of the slope of the line is equal to kl / ml. 4. if you were to graph a line with a slope of 3/2 that intercepts the y - axis at - 2 on the coordinate plane, one point on that line would have a y coordinate of 4 with an x coordinate of what number?

Explanation:

Step1: Recall slope - formula

The slope \(m\) of a line in the coordinate - plane is given by \(m=\frac{\text{rise}}{\text{run}}\). For two right - triangles formed by a line, the ratio of the vertical side to the horizontal side of either triangle is equal to the slope of the line. In right - triangles \(JHI\) and \(MLK\), the absolute value of the slope of the line is \(\frac{ML}{JH}\) (where \(ML\) is the vertical change and \(JH\) is the horizontal change).

Step2: Analyze option A

The absolute value of the slope of the line is the ratio of the length of the vertical side of one of the right - triangles formed by the line to the length of the horizontal side of the same right - triangle. For the right - triangles \(JHI\) and \(MLK\) on the line, the absolute value of the slope \(|m|=\frac{ML}{JH}\), so option A is correct.

Step3: Analyze option B

Since the two right - triangles \(JHI\) and \(MLK\) are similar (because they are formed by the same line and the right - angles are equal), \(\angle L=\angle H\) (corresponding angles of similar triangles), so option B is incorrect.

Step4: Analyze option C

The absolute value of the slope is \(\frac{ML}{JH}\), not \(\frac{IH}{JH}\), so option C is incorrect.

Step5: Analyze option D

The absolute value of the slope is \(\frac{ML}{JH}\), not \(\frac{JH}{ML}\), so option D is incorrect.

Step6: For the second question

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. Given \(m=\frac{3}{2}\) and \(b=-2\), the equation of the line is \(y=\frac{3}{2}x - 2\).
We want to find \(x\) when \(y = 4\). Substitute \(y = 4\) into the equation:
\[

$$\begin{align*} 4&=\frac{3}{2}x-2\\ 4 + 2&=\frac{3}{2}x\\ 6&=\frac{3}{2}x\\ x&=6\times\frac{2}{3}\\ x& = 4 \end{align*}$$

\]

Answer:

1. A. The absolute value of the slope of the line is \(\frac{ML}{JH}\)
2. 4