Show the points M(-2,1),N(5,15),P(6,9),Q(2,1) are vertex in trapezoid and calculate surface mnpq?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to prove that there exists one pair of parallel sides, hence, you need to test if MN is parallel with PQ or MQ is parallel with NP.

You may use slope formula to test the parallelism of sides, such that:

m_(MN) = m_(PQ) => MN || PQ

m_(MQ) = m_(NP) => MQ || NP

You need to evaluate the slopes such that:

m_(MN) = (y_N - y_M)/(x_N - x_M)

m_(MN) = (15 - 1)/(5 - (-2)) => m_(MN) = 14/7 => m_(MN) = 2

m_(PQ) = (y_Q - y_P)/(x_Q - x_P) => m_(PQ) = (1 - 9)/(2 - 6)

m_(PQ) = (-8)/(-4) => m_(PQ) = 2

Since m_(MN) = m_(PQ) = 2, hence MN||PQ, thus MNPQ is trapezoid.

You may evaluate the area of trapezoid as summation of areas of triangles Delta MNP and Delta MPQ, such that:

`S_(Delta MNP) = (1/2)|[(-2,1,1),(5,15,1),(6,9,1)]|`

S_(Delta MNP) = (1/2)|(-30 + 45 + 6 - 90 + 18 - 5)| => S_(Delta MNP) = 28

`S_(Delta MPQ) = (1/2)|[(-2,1,1),(6,9,1),(2,1,1)]|` `S_(Delta MPQ) = (1/2)(-18 + 6 + 2 - 18 + 2 - 6) => S_(Delta MPQ) = 16`

`S_(MNPQ) = S_(Delta MNP) + S_(Delta MPQ)`

`S_(MNPQ) = 28 + 16 => S_(MNPQ) = 44`

Hence, evaluating the area of trapezoid `MNPQ` yields `S_(MNPQ) = 44.`

m_(PQ) = (y_Q - y_P)/(x_Q - x_P)

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