Question

From the diagram, ABCD is a rectangle. The equation of line BC is given by 3y+x=25. Given that the area of rectangle ABCD is 80 units². Find the coordinates of the points B, C and D.​ Point A(-1,2).​

Answer 1

Answer:

B (1, 8)

C (13, 4)

D (11, -2)

Step-by-step explanation:

ABCD is a rectangle, so it has four right angles.  The equation of BC is 3y + x = 25, or in slope-intercept form, y = -⅓ x + ²⁵/₃.

That means the slope of AB is 3.  So the equation of AB in point-slope form is:

y − 2 = 3 (x − (-1))

Or in slope-intercept form:

y − 2 = 3 (x + 1)

y − 2 = 3x + 3

y = 3x + 5

B is the intersection of these two lines.

3x + 5 = -⅓ x + ²⁵/₃

9x + 15 = -x + 25

10x = 10

x = 1

y = 8

The coordinates of B are (1, 8).

The distance between A and B is:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

d = √((1 − (-1))² + (8 − 2)²)

d = √(2² + 6²)

d = √40

The area of the rectangle is 80 square units, so the distance between B and C is:

A = wh

80 = w√40

w = 80 / √40

w = 80√40 / 40

w = 2√40

In other words, the distance between B and C is double the distance between A and B.  We can use distance formula again to find the coordinates of C, or we can use geometry.

If the right triangle formed by hypotenuse AB is a 2×6 triangle, then the right triangle formed by hypotenuse BC is a 4×12 triangle.

So x = 1 + 12 = 13, and y = 8 − 4 = 4.

The coordinates of C are (13, 4).

Similarly, the coordinates of D are:

x = -1 + 12 = 11

y = 2 − 4 = -2

D (11, -2)