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 Probability: 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

(a) Find the probability that Louise’s line has a slope that is greater than 1.

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Answer

To determine the slope of the line from (0, 0) to any point (x, y) on the grid, we use the formula:

slope = rac{y}{x}

For the slope to be greater than 1, we need:

rac{y}{x} > 1 ightarrow y > x

Next, we analyze the points marked with a dot:

(0, 0) - Not applicable (starts from here)
(1, 1) - Slope = 1 (not greater than 1)
(1, 2) - Slope = 2 (greater than 1)
(1, 3) - Slope = 3 (greater than 1)
(2, 2) - Slope = 1 (not greater than 1)
(2, 3) - Slope = 1.5 (greater than 1)
(3, 3) - Slope = 1 (not greater than 1)

Counting the suitable points:

Suitable points with slopes greater than 1: (1, 2), (1, 3), (2, 3)
Total suitable points = 3
Total points = 15 (excluding (0, 0))

Thus, the probability is:

P(slope > 1) = rac{3}{15} = rac{1}{5}

Step 2

(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)?

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Answer

To find the points that are exactly 5 units away from (0, 0), we use the distance formula. The distance d from a point (x, y) to (0, 0) is given by:

d = oot{(x^2 + y^2)}

Setting d = 5:

oot{(x^2 + y^2)} = 5 ightarrow x^2 + y^2 = 25

Now we list the integer coordinate pairs (x, y) that satisfy this equation:

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

From this, the valid points on the grid (given the marked points): (3, 4), (4, 3), (-3, 4), and (-4, 3).

However, only the first two are among the 16 points marked with dots:

Total = 2 points: (3, 4) and (4, 3).

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

(a) 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)?

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