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Lucinda is writing a coordinate proof to show that a diagonal of a parallelogram partitions the parallelogram into two equal areas.

A parallelogram graphed on a coordinate plane. The vertices of rectangle are labeled as K L M and N. The vertex labeled as K lies on begin ordered pair 0 comma 0 end ordered pair. The vertex labeled as L lies on begin ordered pair x comma 2 y end ordered pair. The coordinate of vertex M is left blank. The vertex labeled as N lies on begin ordered pair 3 x comma 0 end ordered pair. A diagonal is drawn between points K and M.



Enter your answers in the boxes to complete Lucinda's proof.

Since KLMN is a parallelogram and a parallelogram's opposite sides are parallel and congruent, the coordinates for M are (4x, 2y).


In △KMN, the length of the base is and the height is . So an expression for the area of △KMN is .


In △KLM, the length of the base is 3x and the height is 2y. So an expression for the area of △KLM is .


Comparing the area of the two triangles that are formed by a diagonal of the parallelogram shows that a diagonal of a parallelogram partitions the parallelogram into two equal areas.

Lucinda is writing a coordinate proof to show that a diagonal of a parallelogram partitions-example-1
Lucinda is writing a coordinate proof to show that a diagonal of a parallelogram partitions-example-1
Lucinda is writing a coordinate proof to show that a diagonal of a parallelogram partitions-example-2
User MattK
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1 Answer

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Answer:

  • ΔKMN: base, 3x; height, 2y; area, 3xy
  • ΔKLM: area, 3xy

Explanation:

ΔKMN

The base length is the length of the horizontal line segment KN. That length is the difference of the x-coordinates: 3x -0 = 3x.

The height is the difference of the y-coordinate of point M and the y-coordinate of horizontal segment KN. That difference is 2y -0 = 2y.

The area is half the product of base and height:

A = (1/2)bh

A = 1/2(3x)(2y) = 3xy

In ΔKMN, the length of the base is 3x and the height is 2y. So an expression for the area of ΔKMN is 3xy.

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ΔKLM

In ΔKLM, the length of the base is 3x and the height is 2y. So an expression for the area of ΔKLM is 3xy.

User Ryszard Czech
by
6.3k points
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