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 Functions: Figure 1 shows a rectangle ABCD.

Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 11 - 2017 - Paper 2

Question 11

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 11 - 2017 - Paper 2

Step 1

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

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Answer

To find the equation of the line through points A and D, we first determine the gradient of line AB. The equation given is $5y + 2x = 10$ .

Rearranging this equation, we get:
$y = -\frac{2}{5}x + 2.$
This indicates that the gradient of line AB is $-\frac{2}{5}$ . Since lines AD and AB are perpendicular, the gradient of line AD is the negative reciprocal of $-\frac{2}{5}$ , which is $\frac{5}{2}$ .

Using point A on the y-axis where $x=0$ and given the y-coordinate of A is 2, the coordinates of A are $(0, 2)$ .
Using the point-slope form of the line equation, we can write the equation of line AD as: $y - 2 = \frac{5}{2}(x - 0).$
Simplifying gives:
$y = \frac{5}{2}x + 2.$
Now, to express this in the form $2y - 5x = 4$ , we can rearrange:

Multiply by 2: $2y = 5x + 4$ .
Rearranging gives: $2y - 5x = 4$ . Thus, we have shown the desired equation.

Step 2

(b) find the area of the rectangle ABCD.

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Answer

To find the area of rectangle ABCD, we first need to find the lengths of sides AB and AD. We already know the coordinates of A as $(0, 2)$ and need to find the coordinates of B.

Using the equation of line AB: $5y + 2x = 10$ , substituting $y$ with $0$ to find B gives:

$5(0) + 2x = 10 \Rightarrow x = 5.$
So, B is at $(5, 0)$ .

Next, we find the length of AB: $AB = 5 - 0 = 5.$
Next, we find the coordinates for D at $(0, 0)$ , as it lies on the y-axis.

Next, we calculate the length of AD: $AD = 2 - 0 = 2.$
Now we can calculate the area of rectangle ABCD using the formula: $Area = AB \times AD = 5 \times 2 = 10.$
Therefore, the area of rectangle ABCD is 10 square units.

Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 11 - 2017 - Paper 2

Worked Solution & Example Answer:Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 11 - 2017 - Paper 2

(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.

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