Question

Which statement is true about quadrilateral ABCD with vertices A(2, 8), B(3, 11), C(4, 8), and D(3, 5)?

Answer 1

The quadrilateral is a rhombus

Explanation:

Given

`A = (2, 8)`

`B = (3, 11)`

`C = (4, 8)`

`D=(3, 5)`

Required

The true statement

Calculate slope (m) using

`m = (y_2 - y_1)/(x_2 - x_1)`

Calculate distance using:

`d= √((x_2 - x_1)^2 + (y_2 -y_1)^2)`

Calculate slope and distance AB

`m_(AB) = (11 - 8)/(3 - 2)`

`m_(AB) = (3)/(1)`

`m_(AB) = 3` -- slope

`d_(AB)= √((3 - 2)^2 + (11 -8)^2)`

`d_(AB)= √(10)` -- distance

Calculate slope and distance BC

`m_(BC) = (8 - 11)/(4 - 3)`

`m_(BC) = (- 3)/(1)`

`m_(BC) = -3` -- slope

`d_(BC) = \sqrt{(4-3)^2+(8-11)^2`

`d_(BC) = √(10)` --- distance

Calculate slope CD

`m_(CD) = (5 - 8)/(3 - 4)`

`m_(CD) = (- 3)/(- 1)`

`m_(CD) = 3` -- slope

`d_(CD) = √((3-4)^2+(5-8)^2)`

`d_(CD) = √(10)` -- distance

Calculate slope DA

`m_(DA) = (8 - 5)/(2 - 3)`

`m_(DA) = (3)/(- 1)`

`m_(DA) = -3` -- slope

`d_(DA) = √((2-3)^2 + (8-5)^2)`

`d_(DA) = √(10)`

From the computations above, we can see that all 4 sides are equal, i.e.
`√(10)`

And the slope of adjacent sides are negative reciprocal, i.e.

`m_(AB) = 3` and
`m_(CD) = -3`

`m_(CD) = 3` and
`m_(DA) = -3`

The quadrilateral is a rhombus