Question

In the given graph ABCD is a parallelogram.

Using the graph, answer the following:

1. write down the coordinates of A, B, C and D.
2. calculate the coordinates of ‘P’, the point of intersection of the diagonals AC and BD.
3. find the slope of sides CB and DA and verify that they represent parallel lines.
4. find the equation of the diagonal AC.

Answer 1

(a) From the graph, the coordinates of the points are:

A = (3, 3),

B = (0, −2),

C = (−4, −2),

D = (−1, 3).

(b) Since point P is the midpoint of both diagonals AC and BD:

For diagonal AC, with A(3, 3) and C(−4, −2), the midpoint is calculated as:

P = `((3 + (−4))/2, (3 + (−2))/2) = (−1/2, 1/2)`

For diagonal BD, with B(0, −2) and D(−1, 3), the midpoint is:

P = `((0 + (−1))/2, (−2 + 3)/2) = (−1/2, 1/2)`

Thus, the diagonals intersect at the same midpoint.

(c) To find the slope of side CB, using points C(−4, −2) and B(0, −2):

`m_1​ = (− 2 + 2)/(0 + 4) = 0`

To find the slope of side DA, using points D(−1, 3) and A(3, 3):

`m_2​ = (3 − 3)/(3 + 1) = 0`

Since m₁ = m_2, sides CB and DA are parallel. Hence verified.

(d) To find the equation of diagonal AC, using points A(3, 3) and C(−4, −2):

First, calculate the slope:

m = `(−2 −3)/(−4 −3) = 5/7`

Using the point-slope form:

`y − 3 = 5/7 ​(x − 3)`

Simplifying, we get:

`y = 5/7​x − 43/7​`

7y − 5x + 43 = 0