To fully answer the given question, let's break down the problem and solve for the x-intercept of the line CD, which is perpendicular to AB and passes through the point C(5, 12).
1. Calculate the Slope of AB:
- The slope formula is [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
- For points A(-10, -3) and B(7, 14):
[tex]\[
\text{slope}_{AB} = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1
\][/tex]
- Therefore, the slope of AB is [tex]\(1.0\)[/tex].
2. Determine the Slope of CD:
- Since CD is perpendicular to AB, the slope of CD is the negative reciprocal of the slope of AB.
- The negative reciprocal of [tex]\(1.0\)[/tex] is [tex]\(-1.0\)[/tex].
- Therefore, the slope of CD is [tex]\(-1.0\)[/tex].
3. Find the Equation of Line CD:
- The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
- We know the line CD passes through point C(5, 12) and has a slope of [tex]\(-1.0\)[/tex].
- Substitute [tex]\(m = -1.0\)[/tex], [tex]\(x = 5\)[/tex], and [tex]\(y = 12\)[/tex] into the equation to solve for [tex]\(b\)[/tex]:
[tex]\[
12 = -1.0 \cdot 5 + b \implies 12 = -5 + b \implies b = 12 + 5 = 17
\][/tex]
- Thus, the y-intercept [tex]\(b\)[/tex] of line CD is [tex]\(17.0\)[/tex].
4. Find the X-intercept of Line CD:
- The x-intercept occurs where [tex]\(y = 0\)[/tex].
- Set the equation of line CD to [tex]\(0\)[/tex] and solve for [tex]\(x\)[/tex]:
[tex]\[
0 = -1.0 \cdot x + 17 \implies -17 = -1.0 \cdot x \implies x = \frac{17}{1} = 17
\][/tex]
- Therefore, the x-intercept of line CD is [tex]\(17.0\)[/tex].
To summarize:
- The slope of AB is [tex]\(1.0\)[/tex],
- The slope of CD is [tex]\(-1.0\)[/tex],
- The y-intercept of CD is [tex]\(17.0\)[/tex],
- The x-intercept of CD is [tex]\(17.0\)[/tex].
Hence, the x-intercept of CD is [tex]\(17.0\)[/tex].
