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9 (a) Triangle A is drawn on the grid - OCR - GCSE Maths - Question 9 - 2017 - Paper 1
Question 9
9 (a) Triangle A is drawn on the grid. Enlarge triangle A with scale factor \( \frac{1}{3} \) and centre of enlargement (\(-1, 5\)). (b) Prism P and prism Q are si... show full transcript
Worked Solution & Example Answer:9 (a) Triangle A is drawn on the grid - OCR - GCSE Maths - Question 9 - 2017 - Paper 1
Step 1
Enlarge triangle A with scale factor \( \frac{1}{3} \) and centre of enlargement (\(-1, 5\))
Answer
To enlarge triangle A, we need to find the new coordinates for each vertex by applying the scale factor and the centre of enlargement. Given the vertices of triangle A at (1, 6), (2, 6), and (2, 4), we will calculate the new coordinates as follows:
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For vertex (1, 6):
New x-coordinate = (-1 + \frac{1}{3} \times (1 + 1) = -1 + \frac{2}{3} = -\frac{1}{3})
New y-coordinate = (5 + \frac{1}{3} \times (6 - 5) = 5 + \frac{1}{3} = \frac{16}{3})
New vertex = ((-\frac{1}{3}, \frac{16}{3})) -
For vertex (2, 6):
New x-coordinate = (-1 + \frac{1}{3} \times (2 + 1) = -1 + 1 = 0)
New y-coordinate = (5 + \frac{1}{3} \times (6 - 5) = 5 + \frac{1}{3} = \frac{16}{3})
New vertex = (0, (\frac{16}{3})) -
For vertex (2, 4):
New x-coordinate = (-1 + \frac{1}{3} \times (2 + 1) = -1 + 1 = 0)
New y-coordinate = (5 + \frac{1}{3} \times (4 - 5) = 5 - \frac{1}{3} = \frac{14}{3})
New vertex = (0, (\frac{14}{3}))
Thus, the new vertices of triangle A after enlargement are: ((-\frac{1}{3}, \frac{16}{3})), (0, (\frac{16}{3})), and (0, (\frac{14}{3})).
Step 2
Explain why he is wrong.
Answer
Jay is incorrect because the relationship between the surface areas and heights of similar prisms does not follow a direct ratio. The surface area ratio is given as 1:3, which implies that the ratio of the heights of the prisms is the square root of the area ratio. Hence, the height ratio is given by:\n[ \text{Height ratio} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \approx 0.577. ]\nTherefore, height of prism P cannot simply be one third of prism Q's height.
Step 3
Calculate the volume of prism P.
Answer
Given that the volume of prism Q is 86 cm³, we can use the relationship between the volumes of similar prisms. The volume ratio is equal to the cube of the height ratio. Thus: [\text{Volume ratio} = \left( \frac{1}{\sqrt{3}} \right)^3 = \frac{1}{3\sqrt{3}}.] This implies that: [ V_P = V_Q \times \frac{1}{3}: = 86 \times \frac{1}{3} = \frac{86}{3} \approx 28.67 : \text{cm³}. ] Therefore, the volume of prism P is approximately 28.67 cm³.