Question

ab is parallel to dc because the gradients are equal.

bc is parallel to ad because the gradients are equal.

to find the lengths of the sides of quadrilateral abcd

click through the slideshow for an example.

ab is the same length as cd.

bc is the same length as ad.

to identify a right angle at vertex

Explanation:

Identify the coordinates of the vertices

We are given the coordinates of the vertices of the quadrilateral \(ABCD\) from the graph:

• \(A(-1, 1)\)
• \(B(5, 4)\)
• \(C(9, -4)\)
• \(D(3, -7)\)

Apply the distance formula for AB and CD

Using Coordinate Geometry, we calculate the lengths of the opposite sides \(AB\) and \(CD\) using the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

For \(AB\) with \(A(-1, 1)\) and \(B(5, 4)\):

$$AB = \sqrt{(5 - (-1))^2 + (4 - 1)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}$$

For \(CD\) with \(C(9, -4)\) and \(D(3, -7)\):

$$CD = \sqrt{(3 - 9)^2 + (-7 - (-4))^2} = \sqrt{(-6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}$$

Apply the distance formula for BC and AD

We calculate the lengths of the other pair of opposite sides \(BC\) and \(AD\):

For \(BC\) with \(B(5, 4)\) and \(C(9, -4)\):

$$BC = \sqrt{(9 - 5)^2 + (-4 - 4)^2} = \sqrt{4^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$$

For \(AD\) with \(A(-1, 1)\) and \(D(3, -7)\):

$$AD = \sqrt{(3 - (-1))^2 + (-7 - 1)^2} = \sqrt{4^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$$

Summarize the side lengths

The lengths of the opposite sides are equal:

• \(AB = CD = 3\sqrt{5} \approx 6.71\)
• \(BC = AD = 4\sqrt{5} \approx 8.94\)

Answer:

The lengths of the sides of quadrilateral \(ABCD\) are:

• \(AB = 3\sqrt{5}\) (approximately \(6.71\))
• \(BC = 4\sqrt{5}\) (approximately \(8.94\))
• \(CD = 3\sqrt{5}\) (approximately \(6.71\))
• \(AD = 4\sqrt{5}\) (approximately \(8.94\))