Math Snap
PROBLEM
Given points , , , and , which slopes confirm diameter bisects chord ?
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 bisects chord .
STEP 2
First, we need to find the slope of the line segment . The slope of a line segment with endpoints and is given by the formula
STEP 3
Now, plug in the given values for the coordinates of points and to calculate the slope of .
STEP 4
Calculate the slope of .
STEP 5
Next, we need to find the slope of the line segment . We can use the same formula as in step2.
STEP 6
Now, plug in the given values for the coordinates of points and $$ to calculate the slope of $\overline{C D}$.
STEP 7
Calculate the slope of .
STEP 8
Recall that two lines are perpendicular if the product of their slopes is . Therefore, if is perpendicular to , then .
STEP 9
Plug in the values for and to check if they are perpendicular.
SOLUTION
Calculate the product of the slopes.
Since the product of the slopes is , is perpendicular to . Therefore, according to the theorem, diameter bisects chord . The key statement in the proof is"The slope of is and the slope of is ."