The co-ordinate diagram below shows part of the town where Cliodhna lives. Each small square in the grid has sides of length 1 cm. (a) Write down the co-ordinates of the Shop, Home, and School. Shop: ( , ) Home: ( , ) School: ( , ) (b) The Library is at the midpoint between Home and the Shop. Write down the co-ordinates of the Library. (c) Work out the distance from Home to the Shop on the diagram. Give your answer in cm, correct to two decimal places. Show your working out. (d) The distance on the diagram from the Shop to the School is roughly 8.1 cm. The diagram is to a scale of 1: 2500. Work out the actual distance from the Shop to the School. Give your answer in metres. (e) Cliodhna walks from the Shop to the School. Explain why the distance she walks is probably more than the actual distance from part (d). (f) The co-ordinate diagram below has the line l drawn from the Shop to the School. The angle A is marked. (g) Work out the slope of the line l. Give your answer as a fraction. (h) Find the equation of the line l. Give your answer in the form ax + by + c = 0, where a, b, c ∈ ℤ. (i) Use trigonometry and the slope of l to work out the size of the angle A. Give your answer in degrees, correct to one decimal place.

Junior Cycle Mathematics Co-Ordinate Geometry: The co-ordinate diagram below sh

The co-ordinate diagram below shows part of the town where Cliodhna lives - Junior Cycle Mathematics - Question 5 - 2019

Question 5

The co-ordinate diagram below shows part of the town where Cliodhna lives. Each small square in the grid has sides of length 1 cm. (a) Write down the co-ordinates o... show full transcript

Worked Solution & Example Answer:The co-ordinate diagram below shows part of the town where Cliodhna lives - Junior Cycle Mathematics - Question 5 - 2019

Step 1

Write down the co-ordinates of the Shop, Home, and School.

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Answer

Shop: (-3, 1) Home: (2, 4) School: (4, 5)

Step 2

Write down the co-ordinates of the Library.

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Answer

The coordinates of the Library, being the midpoint between Home and Shop, can be calculated as follows:

ext{Library} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-3 + 2}{2}, \frac{1 + 4}{2} \right) = \left( -0.5, 2.5 \right)

Thus, the Library is at (-0.5, 2.5).

Step 3

Work out the distance from Home to the Shop on the diagram.

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Answer

To calculate the distance from Home to the Shop, we can use the distance formula:

Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - (-3))^2 + (4 - 1)^2}

Calculating this gives us:

= \sqrt{(5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 ext{ cm}

Step 4

Work out the actual distance from the Shop to the School.

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From the question, we know the distance from the Shop to the School on the diagram is 8.1 cm. Given the scale of 1:2500, we can find the actual distance as follows:

Actual ext{ distance} = 8.1 ext{ cm} \times 2500 = 20250 ext{ cm} = 202.5 ext{ m}

Step 5

Explain why the distance she walks is probably more than the actual distance from part (d).

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Cliodhna likely walks a longer distance due to deviations along the path, which may include following roads or pathways that are not straight lines. The actual distance calculated is direct, but her route may include curves or obstacles that increase the total distance she travels.

Step 6

Work out the slope of the line l.

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Answer

To calculate the slope (m) of line l between the Shop and School, use the formula:

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 1}{4 - (-3)} = \frac{4}{7}

Step 7

Find the equation of the line l.

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To find the equation in the form $ax + by + c = 0$ , we start from the point-slope form:

y - y_1 = m(x - x_1)\ \Rightarrow y - 1 = \frac{4}{7}(x + 3)\ \Rightarrow 7(y - 1) = 4(x + 3)\ \Rightarrow 7y - 4x - 12 - 7 = 0\ \Rightarrow 4x - 7y + 19 = 0

Step 8

Use trigonometry and the slope of l to work out the size of the angle A.

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Answer

To find the angle A, we can use the tangent function, as the slope represents the tangent:

tan(A) = \frac{4}{7}\ A = \tan^{-1}\left( \frac{4}{7} \right) \approx 29.74^{\circ} \approx 29.7^{\circ} \text{ (to one decimal place)}

The co-ordinate diagram below shows part of the town where Cliodhna lives - Junior Cycle Mathematics - Question 5 - 2019

Worked Solution & Example Answer:The co-ordinate diagram below shows part of the town where Cliodhna lives - Junior Cycle Mathematics - Question 5 - 2019

Write down the co-ordinates of the Shop, Home, and School.

Write down the co-ordinates of the Library.

Work out the distance from Home to the Shop on the diagram.

Work out the actual distance from the Shop to the School.

Explain why the distance she walks is probably more than the actual distance from part (d).

Work out the slope of the line l.

Find the equation of the line l.

Use trigonometry and the slope of l to work out the size of the angle A.

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