The points A(2, 1), B(6, 3), C(5, 5), and D(1, 3) are the vertices of the rectangle ABCD as shown - Leaving Cert Mathematics - Question 3 - 2017

To find the area of rectangle ABCD, we can use the formula:

$ext{Area} = ext{Length} imes ext{Width}$

First, we calculate the lengths of sides:

1. Length of AB:

   • B(6, 3) and A(2, 1)
   • Using the distance formula again: $|AB| = ext{sqrt}((6 - 2)^2 + (3 - 1)^2) = ext{sqrt}(16 + 4) = ext{sqrt}(20)$

2. Length of AD:

   • We already calculated this to be √5.

The area can be calculated using the width and height:

i.e., Area = |AB| × |AD| = 4 × 2 = 8 square units (using AB as width and AD as height).

Alternatively, we could also consider the area based on coordinates of rectangle vertices, where:

$ext{Area} = |x_2 - x_1| imes |y_2 - y_1|$

Hence, the area of rectangle ABCD is 10 square units.

To find the equation of line BC, we start by identifying the coordinates of points B(6, 3) and C(5, 5).

1. Calculate the slope (m): $m_{BC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{5 - 6} = \frac{2}{-1} = -2$

2. Using point-slope form ($y - y_1 = m(x - x_1)$): Taking point B(6, 3): $y - 3 = -2(x - 6)$ Rearranging gives: $y = -2x + 12 + 3$ $y = -2x + 15$

3. Writing in the form ax + by + c = 0: $2x + y - 15 = 0$

To find angle ABD, we can apply trigonometry principles. We can use the tangent function, since we have the opposite and adjacent sides.

1. Knowing that:

   • Opposite = |BD| (the length calculated from D(1, 3) to B(6, 3))
   • Adjacent = |AB| (the length calculated from A(2, 1) to B(6, 3)): 4.

2. Calculate |BD|: Using distance formula between B and D: $|BD| = ext{sqrt}((6-1)^2 + (3-3)^2) = ext{sqrt}(25) = 5$

3. Now apply the tangent function: $an(ABD) = \frac{opposite}{adjacent} = \frac{|BD|}{|AB|} = \frac{5}{4}$

4. Using inverse tangent: $heta = an^{-1}(1.25) \approx 51.34^{ ext{o}}$ Thus, angle ABD is approximately 51° when rounded to the nearest degree.