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[tex]$\overleftrightarrow{CD}$[/tex] is perpendicular to [tex]$\overleftrightarrow{AB}$[/tex] and passes through point [tex]$C(5,12)$[/tex].

If the coordinates of [tex]$A$[/tex] and [tex]$B$[/tex] are [tex]$(-10,-3)$[/tex] and [tex]$(7,14)$[/tex], respectively, the [tex]$x$[/tex]-intercept of [tex]$\overleftrightarrow{CD}$[/tex] is [tex]$\square$[/tex]. The point [tex]$\square$[/tex] lies on [tex]$\overleftrightarrow{CD}$[/tex].

Sagot :

GinnyAnswer
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].
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