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Difficulty: MediumGeometric Figures on the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a triangle has vertices at P(−3,1)P(-3, 1), Q(5,1)Q(5, 1), and R(1,7)R(1, 7). A line segment connects the midpoint of side PRPR to the midpoint of side QRQR. What is the length of this line segment?

  1. 4Answer
  2. B
    5
  3. C
    6
  4. D
    8
  5. E
    10

Answer

4
The midpoints of sides PRPR and QRQR are (−1,4)(-1, 4) and (3,4)(3, 4), respectively. Since both midpoints share a y-coordinate of 4, the segment connecting them is horizontal. The length is the difference between their x-coordinates: 3−(−1)=43 - (-1) = 4. Alternatively, by the Triangle Midsegment Theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. The third side, PQPQ, is horizontal and has a length of 5−(−3)=85 - (-3) = 8. Half of this length is 8/2=48 / 2 = 4.

Step-by-Step Solution

1
Find the coordinates of the midpoint of side PRPR.
Midpoint M1=(−3+12,1+72)=(−1,4)M_1 = \left(\frac{-3 + 1}{2}, \frac{1 + 7}{2}\right) = (-1, 4)
The midpoint formula is (x1+x22,y1+y22)\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).
2
Find the coordinates of the midpoint of side QRQR.
Midpoint M2=(5+12,1+72)=(3,4)M_2 = \left(\frac{5 + 1}{2}, \frac{1 + 7}{2}\right) = (3, 4)
Applying the midpoint formula to vertices Q(5,1)Q(5, 1) and R(1,7)R(1, 7).
3
Calculate the distance between the two midpoints M1(−1,4)M_1(-1, 4) and M2(3,4)M_2(3, 4).
Distance =3−(−1)=4= 3 - (-1) = 4
Since the y-coordinates are the same, the distance is simply the absolute difference of the x-coordinates.

Key Concept

Midpoint formula and distance calculations on the coordinate plane, or the application of the Triangle Midsegment Theorem.

Alternative Method

By the Triangle Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. The third side is PQPQ, which has a horizontal length of 5−(−3)=85 - (-3) = 8. Therefore, the midsegment length is 8/2=48 / 2 = 4.
Estimated Time:1m 30s
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Topics

Geometric Figures on the Coordinate Plane

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