Asked by Story
On a coordinate plane, parallelogram P Q R S is shown. Point P is at (5, 1), point Q is at (6, 4), point R is at (3, 10), and point S is at (2, 7).
Which statement proves that PQRS is a parallelogram?
The slopes of SP and RQ are both –2 and SP = RQ = StartRoot 45 EndRoot.
The slopes of RS and QP are both 3 and SP = RQ = StartRoot 45 EndRoot.
The midpoint of RP is (4, 5 and one-half) and the slope of RP is Negative nine-halves.
The midpoint of SQ is (4, 5 and one-half) and SQ = 5.
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The first statement.
Reason: SP slope = (1−7)/(5−2) = −6/3 = −2 and RQ slope = (4−10)/(6−3) = −6/3 = −2, so SP ∥ RQ. Also SP = sqrt[(5−2)^2 + (1−7)^2] = sqrt(9+36) = sqrt45 and RQ = sqrt[(6−3)^2 + (4−10)^2] = sqrt45, so SP = RQ. A quadrilateral with one pair of opposite sides both parallel and equal is a parallelogram. Thus PQRS is a parallelogram.
Reason: SP slope = (1−7)/(5−2) = −6/3 = −2 and RQ slope = (4−10)/(6−3) = −6/3 = −2, so SP ∥ RQ. Also SP = sqrt[(5−2)^2 + (1−7)^2] = sqrt(9+36) = sqrt45 and RQ = sqrt[(6−3)^2 + (4−10)^2] = sqrt45, so SP = RQ. A quadrilateral with one pair of opposite sides both parallel and equal is a parallelogram. Thus PQRS is a parallelogram.
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