QuestionGiven points $A(5,19)$ , $B(17,3)$ , $C(11,1)$ , and $D(20.6,8.2)$ , which slopes confirm diameter $\overline{A B}$ bisects chord $\overline{C D}$ ?

Studdy Solution

STEP 1

Assumptions1. Circle has a diameter $\overline{A B}$ with points $A(5,19)$ and $B(17,3)$. . Circle has a chord $\overline{C D}$ with points $C(11,1)$ and $(20.6,8.)$ .
3. The theorem states, "If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord."
4. We need to find the key statement in the proof that diameter $\overline{A B}$ bisects chord $\overline{C D}$ .

STEP 2

First, we need to find the slope of the line segment $\overline{A B}$ . The slope of a line segment with endpoints $(x1, y1)$ and $(x2, y2)$ is given by the formula $lope = \frac{y2 - y1}{x2 - x1}$

STEP 3

Now, plug in the given values for the coordinates of points $A$ and $B$ to calculate the slope of $\overline{A B}$ .
$lope_{AB} = \frac{3 -19}{17 -5}$

STEP 4

Calculate the slope of $\overline{A B}$ .
$lope_{AB} = \frac{-16}{12} = -\frac{4}{3}$

STEP 5

Next, we need to find the slope of the line segment $\overline{C D}$ . We can use the same formula as in step2.

STEP 6

Now, plug in the given values for the coordinates of points $C$ and $$ to calculate the slope of $\overline{C D}$.
$lope_{CD} = \frac{8.2 -1}{20.6 -11}$

STEP 7

Calculate the slope of $\overline{C D}$ .
$lope_{CD} = \frac{7.2}{9.6} = \frac{3}{4}$

STEP 8

Recall that two lines are perpendicular if the product of their slopes is $-1$ . Therefore, if $\overline{A B}$ is perpendicular to $\overline{C D}$ , then $lope_{AB} \times Slope_{CD} = -1$ .

STEP 9

Plug in the values for $lope_{AB}$ and $lope_{CD}$ to check if they are perpendicular.
$lope_{AB} \times Slope_{CD} = -\frac{4}{3} \times \frac{3}{4}$

STEP 10

Calculate the product of the slopes.
$lope_{AB} \times Slope_{CD} = -$ Since the product of the slopes is $-$ , $\overline{A B}$ is perpendicular to $\overline{C D}$ . Therefore, according to the theorem, diameter $\overline{A B}$ bisects chord $\overline{C D}$ . The key statement in the proof is"The slope of $\overline{A B}$ is $-\frac{4}{3}$ and the slope of $\overline{C D}$ is $\frac{3}{4}$ ."

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