Question

The area of the triangle with vertices $(3, 8), (-4, 2)$ and $(5, 1)$ is $\frac{P}{4}$, then the value of $P$ is

• A. $\frac{61}{2}$
• B. $\frac{2}{61}$
• C. $122$
• D. $\frac{1}{122}$

Hint:

Be meticulous with signs when substituting negative coordinates into the area formula. The absolute value function $|...|$ is crucial because physical area cannot be negative, regardless of the order the vertices are chosen.

Answer 1

The Correct Option is C

To find the area of a triangle given its vertices, we use the formula:

\(\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\)

Where the vertices of the triangle are \((x_1, y_1), (x_2, y_2), (x_3, y_3)\).

Given vertices of the triangle are: \((3, 8), (-4, 2), (5, 1)\).

1. Substitute the values into the formula:

\(\text{Area} = \frac{1}{2} \left| 3(2-1) + (-4)(1-8) + 5(8-2) \right|\)

1. Simplify inside the absolute value:

\(\text{Area} = \frac{1}{2} \left| 3 \times 1 + (-4) \times (-7) + 5 \times 6 \right|\)

1. Calculate each term:

\(= \frac{1}{2} \left| 3 + 28 + 30 \right|\)

1. Simplify further:

\(= \frac{1}{2} \left| 61 \right|\)

1. Calculate final area:

\(= \frac{61}{2}\)

However, the area is given as \(\frac{P}{4}\). Therefore, we equate:

\(\frac{61}{2} = \frac{P}{4}\)

Cross multiply to solve for \(P\):

\(2P = 61 \times 4\)

\(2P = 244\)

\(P = \frac{244}{2} = 122\)

Thus, the value of \(P\) is \(122\).

The correct answer is 122.