To solve this problem, we need to find the equations of the lines AB and CD, and then determine certain points that lie on line CD.
First, let's find the slope of line AB.
The coordinates of A are \((-10, -3)\) and B are \((7, 14)\).
Slope of line AB:
\[ \text{slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - (-3)}{7 - (-10)} = \frac{14 + 3}{7 + 10} = \frac{17}{17} = 1 \]
Since line CD is perpendicular to AB, the slope of line CD is the negative reciprocal of the slope of line AB.
\[ \text{slope of CD} = -\frac{1}{1} = -1 \]
We know the point C(5,12) lies on line CD. We can use the point-slope form to write the equation of line CD:
\[ y - y_1 = m(x - x_1) \]
\[ y - 12 = -1(x - 5) \]
\[ y - 12 = -x + 5 \]
\[ y = -x + 17 \]
Now we need to find the x-intercept of line CD, which occurs when \( y = 0 \):
\[ 0 = -x + 17 \]
\[ x = 17 \]
So, the x-intercept of line CD is \( (17, 0) \). The correct answer from the first drop-down is therefore:
\[ C. (17,0) \]
Next, we need to determine which of the given points lies on line CD. We plug in the x and y values of each point into the equation \( y = -x + 17 \) to see if the point satisfies the equation.
1. For \((-5, 24)\):
\[ y = 24 \]
\[ x = -5 \]
\[ 24 = -(-5) + 17 \]
\[ 24 = 5 + 17 \]
\[ 24 = 22 \] (False)
2. For \((-2, 19)\):
\[ y = 19 \]
\[ x = -2 \]
\[ 19 = -(-2) + 17 \]
\[ 19 = 2 + 17 \]
\[ 19 = 19 \] (True)
3. For \((7, -10)\):
\[ y = -10 \]
\[ x = 7 \]
\[ -10 = -7 + 17 \]
\[ -10 = 10 \] (False)
4. For \((8, 11)\):
\[ y = 11 \]
\[ x = 8 \]
\[ 11 = -8 + 17 \]
\[ 11 = 9 \] (False)
The correct point that lies on line CD is:
\[ B. (-2,19) \]
So, the answers are:
C. (17,0) and B. (-2,19).
Select the correct answer from each drop-down menu.
Line CD is perpendicular to line AB and passes through point C(5,12).
If the coordinates of A and B are (-10,-3) and (7,14), respectively, the x-intercept of line CD is A. (12,0) B. (15,0) C. (17,0) D. (19,0). The point A. (-5,24) B. (-2,19) C. (7,-10) D. (8,11) lies on line CD.
1 answer