A straight line intersects x and y axes at P and Q respectively If (3,5) is the middle point of PQ, then what is the area ofthe triangle OPQ?a)12 square unitsb)15 square unitsc)20 square unitsd)30 square unitsCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

To find the area of triangle OPQ, we need to determine the length of OP and OQ.

Given that (3,5) is the midpoint of PQ, we can use the midpoint formula to find the coordinates of P and Q.

Midpoint formula:
For a line segment with endpoints (x1, y1) and (x2, y2), the coordinates of the midpoint (x, y) are given by:
x = (x1 + x2)/2
y = (y1 + y2)/2

Let's denote the coordinates of P as (xP, 0) and the coordinates of Q as (0, yQ).

Using the midpoint formula, we have:
(xP + 0)/2 = 3 => xP = 6
(0 + yQ)/2 = 5 => yQ = 10

So, the coordinates of P are (6, 0) and the coordinates of Q are (0, 10).

We can now find the lengths of OP and OQ using the distance formula.

Distance formula:
The distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Length of OP:
OP = sqrt((6 - 0)^2 + (0 - 5)^2) = sqrt(36 + 25) = sqrt(61)

Length of OQ:
OQ = sqrt((0 - 0)^2 + (10 - 5)^2) = sqrt(0 + 25) = sqrt(25) = 5

Now, we can calculate the area of triangle OPQ using the formula for the area of a triangle given its base and height.

Area of a triangle:
The area of a triangle with base b and height h is given by:
A = (1/2) * b * h

In this case, the base of the triangle is OP and the height is OQ.

Area of triangle OPQ:
A = (1/2) * OP * OQ = (1/2) * sqrt(61) * 5 = (5/2) * sqrt(61)

To simplify the answer, we can approximate the value of sqrt(61) to a decimal.

Using a calculator, sqrt(61) ≈ 7.81

Therefore, the area of triangle OPQ is approximately:
A ≈ (5/2) * 7.81 = 19.525 ≈ 20 square units

Hence, the correct answer is option C) 20 square units.

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