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The co-ordinate diagram below shows part of the N22 road in County Cork - Junior Cycle Mathematics - Question 3 - 2022
Question 3
The co-ordinate diagram below shows part of the N22 road in County Cork. Two points on the road, P and Q, are marked on the diagram. (a) The point Q has co-ordinat... show full transcript
Worked Solution & Example Answer:The co-ordinate diagram below shows part of the N22 road in County Cork - Junior Cycle Mathematics - Question 3 - 2022
Step 1
Step 2
The equation of the line PQ is: x + 7y = 20 Using this, or otherwise, find the co-ordinates of the point where the line PQ crosses the y-axis.
Answer
To find the y-intercept, we need to set x=0 in the equation:
y = \frac{20}{7}
Thus, the coordinates where the line crosses the y-axis are (0, 2.857) or approximately (0, 2.9).
Step 3
A new road is being built through the point Q (6, 2). On the co-ordinate diagram, it will be a straight line segment which is perpendicular to PQ. Work out the equation of this new road.
Answer
First, we need to determine the slope of line PQ:
The slope of PQ is given by:
( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 3}{6 - (-1)} = -\frac{1}{7} )
The perpendicular slope is the negative reciprocal:
( m_{\perp} = 7 )
Using point-slope form:
( y - 2 = 7(x - 6) )
which simplifies to:
( y = 7x - 42 )
Rearranging gives the equation:
( 7x - y - 42 = 0 )
Step 4
The distance |PQ| on the diagram is 7.1 cm, correct to 1 decimal place. 5 mm on the diagram represents 100 m. Use this to work out the actual distance from P to Q. Give your answer in km.
Answer
First, convert cm to mm:
7.1 cm = 71 mm.
Since 5 mm corresponds to 100 m:
( \text{1 mm} = \frac{100}{5} = 20 \text{ m} )
Thus, 71 mm corresponds to:
( 71 \times 20 = 1420 \text{ m} )
To convert to kilometers:
( \frac{1420}{1000} = 1.42 \text{ km} )