Question

3.

a) find the distance between point a and point c.

b) find the distance between point b and point d.

Explanation:

Step1: Identify coordinates of points

First, we determine the coordinates of the points from the graph.

• For point A: Looking at the graph, it is at (0, 2) since it's on the y - axis (x = 0) and at y = 2.
• For point C: It is at (8, 6) as x = 8 and y = 6.
• For point B: It is at (3, 6) (x = 3, y = 6) (assuming the grid has each square with side length 1, and from the graph, B is 3 units to the right of the y - axis and 6 units up).
• For point D: It is at (5, 2) (x = 5, y = 2) (5 units to the right of the y - axis and 2 units up).

Step2: Use distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Part (a): Distance between A(0,2) and C(8,6)

Step1: Substitute into distance formula

Substitute \(x_1 = 0,y_1 = 2,x_2 = 8,y_2 = 6\) into the distance formula:
\(d=\sqrt{(8 - 0)^2+(6 - 2)^2}=\sqrt{8^2+4^2}=\sqrt{64 + 16}=\sqrt{80}\)
Simplify \(\sqrt{80}\): \(\sqrt{16\times5}=4\sqrt{5}\approx8.94\) (or we can also use the horizontal and vertical differences. The horizontal difference between A and C is \(8-0 = 8\) units, and the vertical difference is \(6 - 2=4\) units. Then, by the Pythagorean theorem, since the line AC forms the hypotenuse of a right triangle with legs 8 and 4, \(d=\sqrt{8^{2}+4^{2}}\) as above. Alternatively, if we notice that AB is horizontal? Wait, no, A is (0,2), C is (8,6). Wait, maybe a better way: count the horizontal and vertical changes. The horizontal change (\(\Delta x\)) is \(8 - 0=8\), vertical change (\(\Delta y\)) is \(6 - 2 = 4\). Then distance \(d=\sqrt{(8)^{2}+(4)^{2}}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}\approx8.94\)

Part (b): Distance between B(3,6) and D(5,2)

Step1: Substitute into distance formula

Substitute \(x_1 = 3,y_1 = 6,x_2 = 5,y_2 = 2\) into the distance formula:
\(d=\sqrt{(5 - 3)^2+(2 - 6)^2}=\sqrt{2^2+(- 4)^2}=\sqrt{4 + 16}=\sqrt{20}\)
Simplify \(\sqrt{20}\): \(\sqrt{4\times5}=2\sqrt{5}\approx4.47\)

Part (a) Answer:

The distance between A and C is \(4\sqrt{5}\) (or approximately 8.94)

Part (b) Answer:

The distance between B and D is \(2\sqrt{5}\) (or approximately 4.47)

Answer:

a) \(4\sqrt{5}\) (or \(\approx8.94\))
b) \(2\sqrt{5}\) (or \(\approx4.47\))