To find the x-intercept of the line CD, which is perpendicular to the line AB and passes through the point C(5, 12), follow these steps:
1. Find the Slope of Line AB:
- Given two points, A (-10, -3) and B (7, 14), we can use the slope formula:
[tex]\[
\text{slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1
\][/tex]
2. Determine the Slope of Line CD:
- Since CD is perpendicular to AB, its slope is the negative reciprocal of the slope of AB. Hence:
[tex]\[
\text{slope of CD} = -\frac{1}{\text{slope of AB}} = -\frac{1}{1} = -1
\][/tex]
3. Write the Equation of Line CD:
- The slope-intercept form of a line's equation is [tex]\(y = mx + b\)[/tex]. We know the slope [tex]\(m = -1\)[/tex] and it passes through point C (5, 12). So we substitute these values into the equation to solve for [tex]\(b\)[/tex]:
[tex]\[
y = -x + b
\][/tex]
- Using the coordinates of point C:
[tex]\[
12 = -5 + b \implies b = 12 + 5 = 17
\][/tex]
- Therefore, the equation of line CD is:
[tex]\[
y = -x + 17
\][/tex]
4. Find the X-Intercept of Line CD:
- The x-intercept occurs where [tex]\(y = 0\)[/tex]. Set [tex]\(y = 0\)[/tex] in the equation of CD:
[tex]\[
0 = -x + 17 \implies x = 17
\][/tex]
So, the x-intercept of CD is [tex]\(17\)[/tex].
