Question

to find the gradients of the sides of quadrilateral abcd

click through the slideshow for an example.

gradient of dc = \frac{-3}{-6} = \frac{1}{2}

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.

Explanation:

Identify coordinates

Using the Coordinate Geometry knowledge point, we identify the vertices of the quadrilateral \(ABCD\) from the graph:

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

Calculate gradient of AB

Using the Gradient of Line knowledge point, the formula is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

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

$$m_{AB} = \frac{4 - 1}{5 - (-1)} = \frac{3}{6} = \frac{1}{2}$$

Calculate gradient of BC

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

$$m_{BC} = \frac{-4 - 4}{9 - 5} = \frac{-8}{4} = -2$$

Calculate gradient of CD

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

$$m_{CD} = \frac{-7 - (-4)}{3 - 9} = \frac{-3}{-6} = \frac{1}{2}$$

Calculate gradient of DA

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

$$m_{DA} = \frac{1 - (-7)}{-1 - 3} = \frac{8}{-4} = -2$$

Answer:

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

• Gradient of \(AB = \frac{1}{2}\)
• Gradient of \(BC = -2\)
• Gradient of \(CD = \frac{1}{2}\)
• Gradient of \(DA = -2\)