Answer :
Let's break down how to determine the [tex]$x$[/tex]-intercept of the line [tex]$\stackrel{\leftrightarrow}{CD}$[/tex], which is perpendicular to line [tex]$\overleftrightarrow{AB}$[/tex] and passes through point [tex]$C(5,12)$[/tex]. Given the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] as [tex]\((-10, -3)\)[/tex] and [tex]\( (7, 14) \)[/tex], respectively:
1. Calculate the slope of line [tex]\( AB \)[/tex]:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the given coordinates:
[tex]\[ \text{slope}_{AB} = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1.0 \][/tex]
2. Determine the slope of line [tex]\( CD \)[/tex]:
Line [tex]\( CD \)[/tex] is perpendicular to line [tex]\( AB \)[/tex]. The slope of a line perpendicular to another is the negative reciprocal of the given line's slope. Therefore, the slope of [tex]\( CD \)[/tex] is:
[tex]\[ \text{slope}_{CD} = -\frac{1}{\text{slope}_{AB}} = -\frac{1}{1.0} = -1.0 \][/tex]
3. Find the equation of line [tex]\( CD \)[/tex]:
The general form of the line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. We can substitute the slope [tex]\((m = -1.0)\)[/tex] and the coordinates of point [tex]\( C \)[/tex] [tex]\((5, 12)\)[/tex] to find [tex]\( b \)[/tex]:
[tex]\[ y = -1.0x + b \][/tex]
Using point [tex]\( C(5, 12) \)[/tex]:
[tex]\[ 12 = -1.0 \cdot 5 + b \][/tex]
[tex]\[ 12 = -5 + b \][/tex]
[tex]\[ b = 12 + 5 = 17.0 \][/tex]
Therefore, the equation of line [tex]\( CD \)[/tex] is:
[tex]\[ y = -1.0x + 17.0 \][/tex]
4. Determine the [tex]\( x \)[/tex]-intercept of line [tex]\( CD \)[/tex]:
The [tex]\( x \)[/tex]-intercept of a line occurs where [tex]\( y = 0 \)[/tex]. Set [tex]\( y \)[/tex] to 0 in the line's equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -1.0x + 17.0 \][/tex]
[tex]\[ 1.0x = 17.0 \][/tex]
[tex]\[ x = 17.0 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercept of [tex]\( \stackrel{\leftrightarrow}{CD} \)[/tex] is [tex]\( 17.0 \)[/tex].
Filled in answers:
1. The [tex]\( x \)[/tex]-intercept of [tex]\( \stackrel{\leftrightarrow}{CD} \)[/tex] is [tex]\(\boxed{17.0}\)[/tex]
2. The point [tex]\(\boxed{(17.0, 0)}\)[/tex] matches the intercept point on the x-axis.
Finally, the complete correct answer should be:
```
The x-intercept of [tex]$\stackrel{\longleftrightarrow}{C D}$[/tex] is 17. The point (17.0, 0).
```
1. Calculate the slope of line [tex]\( AB \)[/tex]:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the given coordinates:
[tex]\[ \text{slope}_{AB} = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1.0 \][/tex]
2. Determine the slope of line [tex]\( CD \)[/tex]:
Line [tex]\( CD \)[/tex] is perpendicular to line [tex]\( AB \)[/tex]. The slope of a line perpendicular to another is the negative reciprocal of the given line's slope. Therefore, the slope of [tex]\( CD \)[/tex] is:
[tex]\[ \text{slope}_{CD} = -\frac{1}{\text{slope}_{AB}} = -\frac{1}{1.0} = -1.0 \][/tex]
3. Find the equation of line [tex]\( CD \)[/tex]:
The general form of the line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. We can substitute the slope [tex]\((m = -1.0)\)[/tex] and the coordinates of point [tex]\( C \)[/tex] [tex]\((5, 12)\)[/tex] to find [tex]\( b \)[/tex]:
[tex]\[ y = -1.0x + b \][/tex]
Using point [tex]\( C(5, 12) \)[/tex]:
[tex]\[ 12 = -1.0 \cdot 5 + b \][/tex]
[tex]\[ 12 = -5 + b \][/tex]
[tex]\[ b = 12 + 5 = 17.0 \][/tex]
Therefore, the equation of line [tex]\( CD \)[/tex] is:
[tex]\[ y = -1.0x + 17.0 \][/tex]
4. Determine the [tex]\( x \)[/tex]-intercept of line [tex]\( CD \)[/tex]:
The [tex]\( x \)[/tex]-intercept of a line occurs where [tex]\( y = 0 \)[/tex]. Set [tex]\( y \)[/tex] to 0 in the line's equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -1.0x + 17.0 \][/tex]
[tex]\[ 1.0x = 17.0 \][/tex]
[tex]\[ x = 17.0 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercept of [tex]\( \stackrel{\leftrightarrow}{CD} \)[/tex] is [tex]\( 17.0 \)[/tex].
Filled in answers:
1. The [tex]\( x \)[/tex]-intercept of [tex]\( \stackrel{\leftrightarrow}{CD} \)[/tex] is [tex]\(\boxed{17.0}\)[/tex]
2. The point [tex]\(\boxed{(17.0, 0)}\)[/tex] matches the intercept point on the x-axis.
Finally, the complete correct answer should be:
```
The x-intercept of [tex]$\stackrel{\longleftrightarrow}{C D}$[/tex] is 17. The point (17.0, 0).
```
