To solve this problem, let's go through the information given and find the necessary values step-by-step:
1. Find the slope of the line AB:
Given points [tex]\( A (-10, -3) \)[/tex] and [tex]\( B (7, 14) \)[/tex]:
The slope [tex]\( m_{AB} \)[/tex] is calculated as:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1.0 \][/tex]
So, the slope of line AB is [tex]\( 1.0 \)[/tex].
2. Find the slope of the line CD which is perpendicular to AB:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the first line.
Therefore, the slope [tex]\( m_{CD} \)[/tex] is:
[tex]\[ m_{CD} = -\frac{1}{m_{AB}} = -\frac{1}{1.0} = -1.0 \][/tex]
3. Find the equation of the line CD:
The line CD passes through point [tex]\( C (5, 12) \)[/tex]. Using the point-slope form of a line, [tex]\( y = mx + b \)[/tex], we can substitute [tex]\( m_{CD} = -1.0 \)[/tex] and point C into the equation to find the y-intercept [tex]\( b_{CD} \)[/tex]:
The equation of line CD is:
[tex]\[ y = -1.0 \cdot x + b_{CD} \][/tex]
Using point [tex]\( C (5, 12) \)[/tex]:
[tex]\[ 12 = -1.0 \cdot 5 + b_{CD} \][/tex]
[tex]\[ 12 = -5 + b_{CD} \][/tex]
[tex]\[ b_{CD} = 12 + 5 = 17.0 \][/tex]
Thus, the equation of line CD is:
[tex]\[ y = -1.0 \cdot x + 17.0 \][/tex]
4. Find the x-intercept of CD:
The x-intercept occurs when [tex]\( y = 0 \)[/tex]. Set [tex]\( y \)[/tex] to 0 in the equation of line CD and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -1.0 \cdot x + 17.0 \][/tex]
[tex]\[ x = \frac{17.0}{1.0} = 17.0 \][/tex]
Therefore, the x-intercept of line CD is [tex]\( 17.0 \)[/tex].
5. Verify that point C (5, 12) lies on line CD:
Since point [tex]\( C \)[/tex] was used to derive the equation of line CD, it indeed lies on line CD. The coordinates of point C are [tex]\( (5, 12) \)[/tex].
Thus, the answers are:
- The x-intercept of [tex]\(\overleftrightarrow{C D}\)[/tex] is [tex]\( 17.0 \)[/tex].
- The point [tex]\((5, 12)\)[/tex] lies on [tex]\(\overleftrightarrow{C D}\)[/tex].
Fill in the blanks accordingly:
- The x-intercept of [tex]\(\overleftrightarrow{C D}\)[/tex] is 17.0.
- The point (5, 12) lies on [tex]\(\overleftrightarrow{C D}\)[/tex].
