Figure 1 shows a rectangle ABCD. The point A lies on the y-axis and the points B and D lie on the x-axis as shown in Figure 1. Given that the straight line through the points A and B has equation $5y + 2x = 10$: (a) show that the straight line through the points A and D has equation $2y - 5x = 4$ (b) find the area of the rectangle ABCD.

A-Level Edexcel Maths Pure Rational Expressions: Figure 1 shows a rectangle ABCD.

Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 7 - 2017 - Paper 2

Question 7

Figure 1 shows a rectangle ABCD. The point A lies on the y-axis and the points B and D lie on the x-axis as shown in Figure 1. Given that the straight line through ... show full transcript

Worked Solution & Example Answer:Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 7 - 2017 - Paper 2

Step 1

show that the straight line through the points A and D has equation $2y - 5x = 4$

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Answer

To show that the equation of the line through points A and D is $2y - 5x = 4$ , we first need to find the gradient of line AB given by the equation $5y + 2x = 10$ .

Rearranging the equation, we have:

$5y = -2x + 10$

$y = -\frac{2}{5}x + 2$

This shows that the gradient of line AB is $m_{AB} = -\frac{2}{5}$ .
Since lines AD and AB are perpendicular, the gradient of line AD, $m_{AD}$ , can be found using the negative reciprocal of $m_{AB}$ :

$m_{AD} = \frac{5}{2}$
Point A is on the y-axis, therefore its coordinates are $(0, y_A)$ , where $y_A$ can be determined by substituting $x = 0$ into the equation of line AB:
$y_A = 2$$ So point A is $(0, 2)$.$
Using the point-slope form of a line with point A and gradient $m_{AD}$ :
$y - 2 = \frac{5}{2}(x - 0)\ y = \frac{5}{2}x + 2$$$
Rearranging gives:

$2y - 5x = 4$

Thus, we have shown that the equation of the line through points A and D is indeed $2y - 5x = 4$ .

Step 2

find the area of the rectangle ABCD

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Answer

To find the area of rectangle ABCD, we need the lengths of sides AB and AD.

Length of AB:
Since A is $(0, 2)$ and B lies on the x-axis, we can find the coordinates of B using the equation of line AB. When $y = 0$ :
$2x = 10\ x = 5$$ Thus, coordinates of B are $(5, 0)$. The length of AB is calculated as: $$AB = |y_A - y_B| = |2 - 0| = 2$$$
Length of AD:
Using Pythagoras' Theorem, where AD is the height from A to the x-axis, we already determined that: $AD = ext{Height} = |y_A| = 2$ Since point D is also on the x-axis:

$AD = ext{distance from A to D} = \sqrt{(5 - 0)^2 + (0 - 2)^2} = \sqrt{25 + 4} = \sqrt{29}$
Area Calculation:
The area of rectangle ABCD is given by:

$ext{Area} = AB \times AD = 2 \times 5 = 10$

Thus, the area of rectangle ABCD is $11.6$ .

Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 7 - 2017 - Paper 2

Worked Solution & Example Answer:Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 7 - 2017 - Paper 2

show that the straight line through the points A and D has equation $2y - 5x = 4$

find the area of the rectangle ABCD

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