Question

points g, h, and k are shown in the xy - coordinate plane.
a. plot a point j so that the line passing through h and j is horizontal.
b. plot a point m so that the line passing through k and m has an undefined slope.
c. write the equation of the line that has the same slope as the line in part (a), but that passes through point g.
d. write the equation of the line that passes through h and g.

Explanation:

Step1: Recall properties of horizontal lines

A horizontal line has a slope of 0 and all points on it have the same y - coordinate. Let the coordinates of point H be \((x_H,y_H)\). We can choose any x - value for point J, say \(x_J=x_H + 1\), and \(y_J = y_H\). For example, if \(H=( - 1,1)\), we can choose \(J=(0,1)\).

Step2: Recall properties of lines with undefined slope

A line with an undefined slope is a vertical line. All points on a vertical line have the same x - coordinate. Let the coordinates of point K be \((x_K,y_K)\). We can choose any y - value for point M, say \(y_M=y_K + 1\), and \(x_M=x_K\). For example, if \(K=(2, - 1)\), we can choose \(M=(2,0)\).

Step3: Find equation of horizontal line through G

A horizontal line has the equation \(y = c\), where c is the y - coordinate of any point on the line. Let the coordinates of point G be \((x_G,y_G)\). The equation of the line with the same slope as the horizontal line in part (a) and passing through G is \(y=y_G\).

Step4: Use two - point form to find line through H and G

The two - point form of a line is \(y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line. Let \(H=(x_H,y_H)\) and \(G=(x_G,y_G)\). The slope \(m=\frac{y_H - y_G}{x_H - x_G}\). Then the equation of the line is \(y - y_H=\frac{y_H - y_G}{x_H - x_G}(x - x_H)\).

Answer:

a. Plot a point J with the same y - coordinate as H and a different x - coordinate.
b. Plot a point M with the same x - coordinate as K and a different y - coordinate.
c. \(y = y_G\) (where \((x_G,y_G)\) are the coordinates of point G)
d. \(y - y_H=\frac{y_H - y_G}{x_H - x_G}(x - x_H)\) (where \((x_H,y_H)\) are the coordinates of point H and \((x_G,y_G)\) are the coordinates of point G)