In the co-ordinate diagram below, 16 points are marked with a dot (•). (a) Louise picks 1 point at random from the 16 points marked with a dot in the diagram. She then finds the equation of the line that goes through this point and through (0, 0). Find the probability that Louise's line has a slope that is greater than 1. (b) How many of the 16 points marked with a dot in the diagram are a distance of exactly 5 units from the point (0, 0)?

Junior Cycle Mathematics Co-Ordinate Geometry: In the co-ordinate diagram below

In the co-ordinate diagram below, 16 points are marked with a dot (•) - Junior Cycle Mathematics - Question 5 - 2021

Question 5

In the co-ordinate diagram below, 16 points are marked with a dot (•). (a) Louise picks 1 point at random from the 16 points marked with a dot in the diagram. She t... show full transcript

Worked Solution & Example Answer:In the co-ordinate diagram below, 16 points are marked with a dot (•) - Junior Cycle Mathematics - Question 5 - 2021

Step 1

Find the points with a slope greater than 1

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Answer

To find the points where the line has a slope greater than 1, we need to look at the coordinates of the points. The slope (m) between two points (x1, y1) and (x2, y2) is given by:

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

In this case, one of the points is (0,0). For the slope to be greater than 1, it must hold that:

$\frac{y_2}{x_2} > 1 \implies y_2 > x_2$

Evaluating each point (1,1), (1,2), (2,1), (2,2), (2,3), (3,2), (3,3), (3,4), (4,3), (4,4), we find the valid points where y > x:

(1, 2)
(2, 3)
(3, 4)

Thus, there are 3 points that yield a slope greater than 1.

The probability is then calculated as follows:

$P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{16}$

Step 2

Find the number of points at a distance of exactly 5 units

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Answer

To determine how many points are exactly 5 units from the point (0, 0), we use the distance formula:

$d = \sqrt{(x - x_1)^2 + (y - y_1)^2}$

Setting d to 5 and (x_1, y_1) to (0, 0), gives:

$5 = \sqrt{x^2 + y^2} \implies 25 = x^2 + y^2$

Now we evaluate integer solutions (x, y) for the equation x^2 + y^2 = 25 within the range of coordinates from the diagram. The valid points are:

(3, 4)
(4, 3)
(-3, 4)
(-4, 3)
(3, -4)
(4, -3)
(-3, -4)
(-4, -3)

However, from the 16 points in the diagram, only (3, 4) and (4, 3) are visible giving a total count of 2 points.

In the co-ordinate diagram below, 16 points are marked with a dot (•) - Junior Cycle Mathematics - Question 5 - 2021

Worked Solution & Example Answer:In the co-ordinate diagram below, 16 points are marked with a dot (•) - Junior Cycle Mathematics - Question 5 - 2021

Find the points with a slope greater than 1

Find the number of points at a distance of exactly 5 units

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