Question

Given: quadrilateral MATH; M(-5,-2), A(-3,2),

T(3,2) and H(1,-2)
Prove: MATH is a parallelogram

Answer 1

If both of the opposite sides are parallel then the quadrilateral is a parallelogram. If the slopes are the same and the constants are different than the lines are parallel.

Explanation:

If both of the opposite sides are parallel then the quadrilateral is a parallelogram. If the slopes are the same and the constants are different than the lines are parallel.

(-32) ( 3,2) and side (-5,-2) (1,-2)

(-3, 2) (3,2)

Slope
`(2-2)/(3- -3)` =
`(0)/(6)` = 0

y-intercept

2 = 0(3) + b

2 = b

Equation

y = 0x + 2

y = 2

(-5,-2) ( 1,-2)

slope
`(-2 - -2)/(1 - -5)` =
`(0)/(6)` = 0

y-intercept

-2 = 0(1) + b

-2 = b

Equation

y = 0x - 2

y = -2

These sides are parallel because the slope is the same and the y-intercepts are different.

Sides (-3,2) (-5,-2) and (3,2) (1,-2)

(-3,2) (-5,-2)

slope
`(-2 -2)/(-5 - -3)` =
`(-4)/(-2)` = 2

y-intercept

-2 = (2)(-5) + b

-2 = -10 + b

8 = b

Equation:

y = 2x + 8

(3,2) (1,-2)

slope
`(-2-2)/(1-3)` =
`(-4)/(-2)` = 2

y-intercept

2 = (2)(3) + b

2 = 6 + b

-4 = b

Equation

y = 2x -4

The sides are parallel because the slopes are the same, but the y-intercept are different.

Answer 2

Given:

• Quadrilateral MATH with vertices M(-5,-2), A(-3,2), T(3,2) and H(1,-2)

Recall properties of a parallelogram:

• Opposite sides are parallel, and congruent;
• Diagonals bisect each-other.

Prove that same applies to the given quadrilateral.

1) Opposite sides are: MH & AT and MA & TH.

MH and AT have same y-coordinates and therefore both are horizontal, hence parallel.

Their lengths:

• MH = 1 - (-5) = 6 units,
• AT = 3 - (-3) = 6 units.

Same can be proved for MA &TH.

2) Find the midpoint of diagonals MT and AH.

Midpoint of MT:

• x = (-5 + 3)/2 = - 2/2 = - 1,
• y = (-2 + 2)/2 = 0/2 = 0.

Midpoint of AH:

• x = (-3 + 1)/2 = - 2/2 = - 1,
• y = (2 - 2)/2 = 0/2 = 0.

Both diagonals have same midpoint (-1, 0). This proves that diagonals bisect each-other.

Therefore MATH is a parallelogram.