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PROBLEM

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

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 AB\overline{A B} bisects chord CD\overline{C D}.

STEP 2

First, we need to find the slope of the line segment AB\overline{A B}. The slope of a line segment with endpoints (x1,y1)(x1, y1) and (x2,y2)(x2, y2) is given by the formulalope=y2y1x2x1lope = \frac{y2 - y1}{x2 - x1}

STEP 3

Now, plug in the given values for the coordinates of points AA and BB to calculate the slope of AB\overline{A B}.
lopeAB=319175lope_{AB} = \frac{3 -19}{17 -5}

STEP 4

Calculate the slope of AB\overline{A B}.
lopeAB=1612=43lope_{AB} = \frac{-16}{12} = -\frac{4}{3}

STEP 5

Next, we need to find the slope of the line segment CD\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 CC and $$ to calculate the slope of $\overline{C D}$.
lopeCD=8.2120.611lope_{CD} = \frac{8.2 -1}{20.6 -11}

STEP 7

Calculate the slope of CD\overline{C D}.
lopeCD=7.29.6=34lope_{CD} = \frac{7.2}{9.6} = \frac{3}{4}

STEP 8

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

STEP 9

Plug in the values for lopeABlope_{AB} and lopeCDlope_{CD} to check if they are perpendicular.
lopeAB×SlopeCD=43×34lope_{AB} \times Slope_{CD} = -\frac{4}{3} \times \frac{3}{4}

SOLUTION

Calculate the product of the slopes.
lopeAB×SlopeCD=lope_{AB} \times Slope_{CD} = -Since the product of the slopes is -, AB\overline{A B} is perpendicular to CD\overline{C D}. Therefore, according to the theorem, diameter AB\overline{A B} bisects chord CD\overline{C D}. The key statement in the proof is"The slope of AB\overline{A B} is 43-\frac{4}{3} and the slope of CD\overline{C D} is 34\frac{3}{4}."

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