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A rhombus is drawn on a coordinate grid - OCR - GCSE Maths - Question 19 - 2023 - Paper 5

Question 19

A rhombus is drawn on a coordinate grid. One diagonal of the rhombus has equation $y = \frac{1}{2}x + 3$. The other diagonal passes through the point (1, 7). Find... show full transcript

Worked Solution & Example Answer:A rhombus is drawn on a coordinate grid - OCR - GCSE Maths - Question 19 - 2023 - Paper 5

Step 1

Find the gradient of the given diagonal

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Answer

The equation of the first diagonal is given as (y = \frac{1}{2}x + 3). Therefore, the gradient (m) of the first diagonal is (m = \frac{1}{2}). The gradient of the second diagonal of a rhombus is the negative reciprocal of the first diagonal's gradient. Thus, the gradient (m_2) of the second diagonal is given by:

$m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{2}} = -2$

Step 2

Use the point-slope form to find the equation

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Answer

Given that the second diagonal passes through the point (1, 7) and has a gradient of (-2), we can use the point-slope form of the equation:

$y - y_1 = m(x - x_1)$

Substituting (x_1 = 1), (y_1 = 7), and (m = -2), we get:

$y - 7 = -2(x - 1)$

Expanding this, we find:

$y - 7 = -2x + 2$

Therefore, simplifying gives:

$y = -2x + 9$

Step 3

Final equation of the second diagonal

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Answer

Thus, the equation of the other diagonal of the rhombus can be expressed in slope-intercept form as:

$y = -2x + 9$

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