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 Geometric Sequences & Series: Figure 1 shows a rectangle ABCD.

Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 10 - 2017 - Paper 2

Question 10

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 10 - 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 find the gradient of the line AB, we start with the equation given:

$5y + 2x = 10$

Rearranging gives:

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

Thus, the gradient of the line AB is $m = -\frac{2}{5}$ . The line AD should have a gradient that is the negative reciprocal of this, thus:

$m_{AD} = \frac{5}{2}$

Since point A lies on the y-axis, its coordinates are $(0, y_A)$ . We know from the equation of line AB that when $x=0$ , $y_A = 2$ . Therefore, the coordinates of A are $(0, 2)$ .

Using the point-slope form of a line's equation, we have:

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

Substituting for A and the derived gradient:

$y - 2 = \frac{5}{2}(x - 0)$

This simplifies to:

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

To rewrite this in standard form:

$2y - 5x = 4$

Hence, we have shown that the straight line through the points A and D has the required equation.

Step 2

find the area of the rectangle ABCD.

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Answer

To find the area of rectangle ABCD, we need to calculate the lengths of sides AB and AD using the coordinates of points A, B, and D.

The coordinates of points are:

A(0, 2)
B(5, 0)
D(0, 0)

Calculating the length AB:

$AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(5 - 0)^2 + (0 - 2)^2} = \sqrt{25 + 4} = \sqrt{29}$

Calculating the length AD:

$AD = y_A - y_D = 2 - 0 = 2$

Now, we can find the area of rectangle ABCD:

$Area = AB \times AD = \sqrt{29} \times 2$

Numerically, the area is approximately:

$Area \approx 2 \times 5.385 = 10.77$

Thus, the area of rectangle ABCD is approximately 11.6.

Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 10 - 2017 - Paper 2

Worked Solution & Example Answer:Figure 1 shows a rectangle ABCD - Edexcel - A-Level Maths Pure - Question 10 - 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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