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The quadrilateral ABCD is shown in the co-ordinate diagram below - Junior Cycle Mathematics - Question 11 - 2021
Question 11
The quadrilateral ABCD is shown in the co-ordinate diagram below. (a) Complete the table below to show the co-ordinates of the four corners of ABCD. Point A ... show full transcript
Worked Solution & Example Answer:The quadrilateral ABCD is shown in the co-ordinate diagram below - Junior Cycle Mathematics - Question 11 - 2021
Step 1
Complete the table below to show the co-ordinates of the four corners of ABCD.
Answer
To find the coordinates of points C and D:
- The point C is located at (8, 0) since it is horizontally aligned with point B (2, 0) and vertically on the line y = 0.
- The point D is at (8, 4), directly above point C at the line y = 4.
Thus, the completed table would be:
Point A B C D Co-ordinates (2,4) (2,0) (8,0) (8,4)
Step 2
On the diagram above, draw the image of ABCD under axial symmetry in the x-axis.
Answer
For axial symmetry in the x-axis, each point (x, y) transforms to (x, -y).
Therefore:
- Point A (2, 4) becomes A' (2, -4)
- Point B (2, 0) becomes B' (2, 0)
- Point C (8, 0) becomes C' (8, 0)
- Point D (8, 4) becomes D' (8, -4).
You would plot these new points A', B', C', and D' in the x-axis reflected position.
Step 3
Work out the area of the shape ABCD.
Answer
To find the area of quadrilateral ABCD, we can split it into a rectangle and a triangle.
-
Calculate the area of Rectangle ABCD:
- Base (AB) = 2 (from point A to B in the x-direction)
- Height (AD) = 4 (from point A to D in the y-direction)
Area of rectangle = base × height = 2 × 4 = 8.
-
Calculate the area of Triangle BCD:
- Base (BC) = 6 (from point B (2, 0) to C (8, 0))
- Height = 4 (from point D (8, 4) straight down to line BC)
Area of triangle = 0.5 × base × height = 0.5 × 6 × 4 = 12.
Total Area = Area of Rectangle + Area of Triangle = 8 + 12 = 20.
Step 4
Write each line segment from the list above into the correct place in the table below, to match each line segment to its equation.
Answer
Based on the coordinates:
- Line Segment [AB] corresponds to the equation y = 4 since points A and D are at y = 4.
- Line Segment [CD] must correspond to the equation y = x - 7, which represents the diagonal line connecting C and D.
- Line Segment [AD] corresponds to the equation y = 0 as it is a horizontal line along y = 0
Final Table:
Equation Line segment x = 2 [AD] y = 0 [BC] y = 4 [AB] y = x - 7 [CD]