Question

select the correct answer from each drop - down menu. cd is perpendicular to 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 cd is (17, 0). the point (-5, 24), (-2, 19), (7, -10), (8, 11) lies on cd.

Explanation:

Step1: Find slope of AB

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $A(-10,-3)$ and $B(7,14)$, $m_{AB}=\frac{14 - (-3)}{7-(-10)}=\frac{14 + 3}{7 + 10}=\frac{17}{17}=1$.

Step2: Find slope of CD

Since $CD$ is perpendicular to $AB$, the product of their slopes is - 1. Let the slope of $CD$ be $m_{CD}$. Then $m_{AB}\times m_{CD}=-1$. Since $m_{AB} = 1$, $m_{CD}=-1$.

Step3: Find equation of CD

The point - slope form of a line is $y - y_1=m(x - x_1)$. Using point $C(5,12)$ and $m=-1$, we have $y - 12=-1(x - 5)$, which simplifies to $y-12=-x + 5$ or $y=-x+17$.

Step4: Find x - intercept of CD

To find the x - intercept, set $y = 0$. Then $0=-x + 17$, so $x = 17$. The x - intercept is the point $(17,0)$.

Step5: Check which point lies on CD

Substitute the x and y values of each point into the equation $y=-x + 17$.
For point $(8,11)$: $y=-x + 17$, when $x = 8$, $y=-8 + 17=9
eq11$.
For point $(7,-10)$: when $x = 7$, $y=-7 + 17 = 10
eq-10$.
For point $(-2,19)$: when $x=-2$, $y=-(-2)+17=19$.
For point $(-5,24)$: when $x=-5$, $y=-(-5)+17=22
eq24$.

Answer:

The x - intercept of $\overleftrightarrow{CD}$ is $(17,0)$. The point $(-2,19)$ lies on $\overleftrightarrow{CD}$.