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Question

Difficulty: MediumDistance and Midpoint Formulas

A circle in the standard (x,y)(x, y) coordinate plane has a diameter with one endpoint at (−3,5)(-3, 5) and its center at (2,−7)(2, -7). What is the yy-coordinate of the other endpoint of the diameter?

Answer: -19

Answer

The y-coordinate of the other endpoint of the diameter is -19.
Because the center of a circle is the midpoint of any diameter, the midpoint formula applies. For the y-coordinate, the equation is ym=y1+y22y_m = \frac{y_1 + y_2}{2}. Substituting the y-coordinate of the given endpoint (55) and the center (−7-7) yields −7=5+y22-7 = \frac{5 + y_2}{2}. Multiplying both sides by 2 gives −14=5+y2-14 = 5 + y_2. Subtracting 5 from both sides results in −19-19.

Step-by-Step Solution

1
Identify the relationship between the circle's center and its diameter.
The center of the circle, (2,−7)(2, -7), is the midpoint of the diameter.
By definition, the center of a circle bisects any diameter, making it the midpoint of the diameter's endpoints.
2
Apply the midpoint formula to set up an equation for the y-coordinate.
ym=y1+y22y_m = \frac{y_1 + y_2}{2}
The y-coordinate of a midpoint is the average of the y-coordinates of the two endpoints.
3
Substitute the given values into the formula and solve for the unknown y-coordinate.
y2=−19y_2 = -19
Substituting the given y-coordinates yields −7=5+y22-7 = \frac{5 + y_2}{2}. Multiplying by 2 gives −14=5+y2-14 = 5 + y_2, and subtracting 5 from both sides yields −19-19.

Key Concept

Finding a missing endpoint given the midpoint and one endpoint
Estimated Time:1m 0s
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Topics

Distance and Midpoint Formulas

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