Photo AI

The co-ordinate diagram below shows part of the town where Ben lives - Junior Cycle Mathematics - Question 10 - 2019

Question icon

Question 10

The-co-ordinate-diagram-below-shows-part-of-the-town-where-Ben-lives-Junior Cycle Mathematics-Question 10-2019.png

The co-ordinate diagram below shows part of the town where Ben lives. (a) Ben's bike is half way between the Shop and the School (that is, the midpoint). Plot a poi... show full transcript

Worked Solution & Example Answer:The co-ordinate diagram below shows part of the town where Ben lives - Junior Cycle Mathematics - Question 10 - 2019

Step 1

Plot a point on the diagram to show where Ben's bike is. Label this point B.

96%

114 rated

Answer

To find the midpoint between the points representing the Shop and the School:

  • The coordinates of the Shop are (-3, 1) and the School are (2, 4).
  • Midpoint formula:
B=(x1+x22,y1+y22)B = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
  • Substituting the values:
B=(−3+22,1+42)=(−0.5,2.5)B = \left( \frac{-3 + 2}{2}, \frac{1 + 4}{2} \right) = \left( -0.5, 2.5 \right)
  • Therefore, the point B is plotted at (-0.5, 2.5).

Step 2

Work out the distance from Home to the Shop on the diagram. Give your answer correct to one decimal place. Show your working out.

99%

104 rated

Answer

To find the distance from Home to the Shop using the distance formula:

  • The coordinates are:
    • Home: (0, 3)
    • Shop: (-3, 1)
  • Distance formula:
D=(x2−x1)2+(y2−y1)2D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  • Substituting the values:
D=((−3)−0)2+(1−3)2=(−3)2+(−2)2=9+4=13≈3.6D = \sqrt{((-3) - 0)^2 + (1 - 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.6
  • Therefore, the distance from Home to the Shop is 3.6 cm.

Step 3

Work out the actual distance from the Shop to the School. Give your answer in metres.

96%

101 rated

Answer

From the previous calculation, we have determined the distance from Shop to School in cm:

  • Distance on the diagram from the Shop to the School is approximated at 5 cm.

  • Given that the scale is 1 : 2500:

  • Actual distance:

Actual distance=Distance on diagram×2500=5 cm×2500=12500 cm\text{Actual distance} = \text{Distance on diagram} \times 2500 = 5 \text{ cm} \times 2500 = 12500 \text{ cm}
  • Converting cm to m:
12500 cm=12500/100=125 m12500 \text{ cm} = 12500 / 100 = 125 \text{ m}
  • Therefore, the actual distance from the Shop to the School is 125 m.

Step 4

Show that the slope of the line from the Shop to Home is \( \frac{1}{3} \).

98%

120 rated

Answer

To find the slope (m) of the line from the Shop to Home:

  • Coordinates:
    • Shop: (-3, 1)
    • Home: (0, 3)
  • Slope formula:
m=riserun=y2−y1x2−x1m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
  • Substituting the values:
m=3−10−(−3)=23=13m = \frac{3 - 1}{0 - (-3)} = \frac{2}{3} = \frac{1}{3}
  • Therefore, the slope is ( \frac{1}{3} ).

Step 5

Use this fact to work out the size of the angle H, correct to the nearest degree.

97%

117 rated

Answer

To find the angle H:

  • Given: ( \tan H = \frac{1}{3} )
  • Using the arctangent function to find the angle:
H=tan⁡−1(13)H = \tan^{-1}\left(\frac{1}{3}\right)
  • Calculating this value:
H≈18.43°H \approx 18.43°
  • Therefore, rounding to the nearest degree, ( H \approx 18° ).

Join the Junior Cycle students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;