Question

The vertices of a parallelogram PQRS are P(a,7),Q(7,5),R(0,4)and S(6,2). Calculate the value of a

Answer 1

To find the value of "a", we need to use the fact that opposite sides of a parallelogram are equal in length.

First, we calculate the distance between points P and Q using the distance formula:

d(PQ) = √[(7-a)^2 + (5-7)^2]
d(PQ) = √[(7-a)^2 + 4]
d(PQ) = √[49 - 14a + a^2 + 4]
d(PQ) = √[a^2 - 14a + 53]

Next, we calculate the distance between points R and S using the distance formula:

d(RS) = √[(6-0)^2 + (2-4)^2]
d(RS) = √[36 + 4]
d(RS) = √40

Since opposite sides of a parallelogram are equal in length, we can set the distances equal to each other:

√[a^2 - 14a + 53] = √40

Squaring both sides to eliminate the square root, we get:

a^2 - 14a + 53 = 40

Rearranging the equation, we get:

a^2 - 14a + 13 = 0

Now, we can solve for "a" by factoring or using the quadratic formula:

(a - 1)(a - 13) = 0

This gives us two possible values for "a": a = 1 or a = 13.

Therefore, the value of "a" can be either 1 or 13.