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Question

Difficulty: MediumDistance and Midpoint Formulas

Points K(−2,5)K(-2, 5) and L(6,−1)L(6, -1) are the endpoints of segment KLKL on a coordinate plane. What is the distance from the midpoint of segment KLKL to the point P(5,7)P(5, 7)?

  1. A
    10\sqrt{10}
  2. 34\sqrt{34}Answer
  3. C
    8
  4. D
    10
  5. E
    130\sqrt{130}

Answer

The correct answer is the square root of 34.
To find the distance from the midpoint of segment KLKL to the point P(5,7)P(5, 7), we first calculate the midpoint MM of KLKL. Using the midpoint formula, the coordinates are x=−2+62=2x = \frac{-2 + 6}{2} = 2 and y=5+(−1)2=2y = \frac{5 + (-1)}{2} = 2, giving M(2,2)M(2, 2). Next, we apply the distance formula to find the distance between M(2,2)M(2, 2) and P(5,7)P(5, 7), which is d=(5−2)2+(7−2)2=32+52=9+25=34d = \sqrt{(5 - 2)^2 + (7 - 2)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}. This confirms the correct distance is the square root of 34.

Step-by-Step Solution

1
Calculate the coordinates of the midpoint of segment KLKL using the midpoint formula: M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).
The midpoint is M(2,2)M(2, 2) since x=−2+62=2x = \frac{-2 + 6}{2} = 2 and y=5+(−1)2=2y = \frac{5 + (-1)}{2} = 2.
Finding the midpoint of the segment is required before finding the distance from it to point PP.
2
Calculate the distance between the midpoint M(2,2)M(2, 2) and the point P(5,7)P(5, 7) using the distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
The distance is 34\sqrt{34} since d=(5−2)2+(7−2)2=32+52=9+25=34d = \sqrt{(5 - 2)^2 + (7 - 2)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}.
This determines the final distance required by the question.

Key Concept

Applying both the midpoint and distance formulas sequentially to solve coordinate geometry problems.
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Topics

Distance and Midpoint Formulas

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