Question

example 3 (pp. 7–8) draw and label a figure for each relationship. 5. a line in a coordinate plane contains x(3, −1), y(−3, −4), and z(−1, −3) and a point w that does not lie on \overleftrightarrow{xy}. 6. plane q contains lines r and s that intersect in p.

Explanation:

Problem 5

Step 1: Plot Points X, Y, Z

First, plot the points \( X(3, -1) \), \( Y(-3, -4) \), and \( Z(-1, -3) \) on a coordinate plane. To check if \( Z \) is on \( \overleftrightarrow{XY} \), we can find the equation of the line through \( X \) and \( Y \). The slope \( m \) of \( XY \) is \( m=\frac{-4 - (-1)}{-3 - 3}=\frac{-3}{-6}=\frac{1}{2} \). Using point - slope form \( y - y_1=m(x - x_1) \) with \( X(3,-1) \), we get \( y+1=\frac{1}{2}(x - 3) \), or \( y=\frac{1}{2}x-\frac{3}{2}-1=\frac{1}{2}x-\frac{5}{2} \). Now, substitute \( x = - 1 \) into the equation: \( y=\frac{1}{2}(-1)-\frac{5}{2}=\frac{-1 - 5}{2}=-3 \), so \( Z(-1,-3) \) is on \( \overleftrightarrow{XY} \).

Step 2: Choose a Point W

Choose a point \( W \) that does not satisfy the equation of \( \overleftrightarrow{XY} \). For example, let's take \( W(0,0) \). Substitute \( x = 0 \) into \( y=\frac{1}{2}x-\frac{5}{2} \), we get \( y=-\frac{5}{2}
eq0 \), so \( W(0,0) \) does not lie on \( \overleftrightarrow{XY} \).

Step 3: Draw the Figure

Draw the coordinate plane. Plot points \( X(3,-1) \), \( Y(-3,-4) \), \( Z(-1,-3) \) (with a line through \( X \) and \( Y \) passing through \( Z \)) and \( W(0,0) \) (marked as a point not on the line \( \overleftrightarrow{XY} \)). Label all the points and the line \( \overleftrightarrow{XY} \).

Problem 6

Step 1: Draw Plane Q

Draw a plane (we can represent a plane as a parallelogram - shaped figure) and label it as Plane \( Q \).

Step 2: Draw Lines r and s

Draw two lines \( r \) and \( s \) inside Plane \( Q \) such that they intersect at a point. Label the point of intersection as \( P \). Also, label the lines as \( r \) and \( s \) and the plane as \( Q \).

(Note: Since these are drawing - based problems, the above steps describe the process of creating the figures. For the final answer, we can describe the figures as follows)

Answer:

(Problem 5):

• A coordinate plane with a line \( \overleftrightarrow{XY} \) passing through \( X(3,-1) \), \( Y(-3,-4) \), and \( Z(-1,-3) \), and a point \( W \) (e.g., \( W(0,0) \)) not on \( \overleftrightarrow{XY} \), all labeled appropriately.