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9 (a) Triangle A is drawn on the grid - OCR - GCSE Maths - Question 9 - 2017 - Paper 1

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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 similar. T... 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)

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Answer

To enlarge triangle A, we calculate the new coordinates using the formula for enlargement:

extNewCoordinate=extCentre+k×(extOldCoordinate−extCentre) ext{New Coordinate} = ext{Centre} + k \times ( ext{Old Coordinate} - ext{Centre})

where:

  • The centre of enlargement is (–1, 5)
  • The scale factor kk is 13\frac{1}{3}.

Assuming triangulations of triangle A are at points (1, 6), (2, 6), and (2, 4):

  1. For point (1, 6):
    extNewx=−1+13((1)−(−1))=−1+13(2)=−1+23=−13ext{New x} = -1 + \frac{1}{3}((1) - (-1)) = -1 + \frac{1}{3}(2) = -1 + \frac{2}{3} = -\frac{1}{3}
    extNewy=5+13((6)−(5))=5+13(1)=5+13=163ext{New y} = 5 + \frac{1}{3}((6) - (5)) = 5 + \frac{1}{3}(1) = 5 + \frac{1}{3} = \frac{16}{3}

  2. For point (2, 6):
    extNewx=−1+13((2)−(−1))=−1+13(3)=−1+1=0ext{New x} = -1 + \frac{1}{3}((2) - (-1)) = -1 + \frac{1}{3}(3) = -1 + 1 = 0
    extNewy=5+13((6)−(5))=5+13(1)=5+13=163ext{New y} = 5 + \frac{1}{3}((6) - (5)) = 5 + \frac{1}{3}(1) = 5 + \frac{1}{3} = \frac{16}{3}

  3. For point (2, 4):
    extNewx=−1+13((2)−(−1))=−1+13(3)=−1+1=0ext{New x} = -1 + \frac{1}{3}((2) - (-1)) = -1 + \frac{1}{3}(3) = -1 + 1 = 0
    extNewy=5+13((4)−(5))=5+13(−1)=5−13=143ext{New y} = 5 + \frac{1}{3}((4) - (5)) = 5 + \frac{1}{3}(-1) = 5 - \frac{1}{3} = \frac{14}{3}

Therefore, the new vertices of the enlarged triangle A are at points (−13,163),(0,163),(0,143)(-\frac{1}{3}, \frac{16}{3}), (0, \frac{16}{3}), (0, \frac{14}{3}).

Step 2

Explain why Jay is wrong

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Answer

Jay's assertion is incorrect because the relationship between the surface area and height of similar prisms does not follow a linear ratio; rather, it follows the square of the height ratio. The height factor is the square root of the area ratio. Given that the ratio of the surface areas of the prisms is 1:3, the height factor should be:

Height factor=13=13≈0.577\text{Height factor} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \approx 0.577

Thus, the height of prism P is not merely one third of the height of prism Q but is scaled by a factor of about 0.5770.577.

Step 3

Calculate the volume of prism P

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Answer

Given the volume of prism Q is 86 cm³ and noting that volume scales with the cube of the height factor:

Volume Factor=(13)3=133≈0.192\text{Volume Factor} = \left(\frac{1}{\sqrt{3}}\right)^3 = \frac{1}{3\sqrt{3}} \approx 0.192

Therefore, the volume of prism P can be calculated as:

Volume of prism P=Volume of prism Q×Volume Factor=86×(13)3=86×133≈86×0.192≈16.55 cm³\text{Volume of prism P} = \text{Volume of prism Q} \times \text{Volume Factor} = 86 \times \left(\frac{1}{\sqrt{3}}\right)^3 = 86 \times \frac{1}{3\sqrt{3}} \approx 86 \times 0.192 \approx 16.55 \text{ cm³}

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