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

Step 1: Understanding the Concept:
We are given the coordinates of the three vertices of a triangle and its area in terms of an unknown variable $P$. We need to compute the actual numerical area using a standard formula and equate it to the given expression to solve for $P$.

Step 2: Key Formula or Approach:
The area of a triangle defined by vertices $(x_1, y_1), (x_2, y_2)$, and $(x_3, y_3)$ can be calculated using the coordinate geometry formula: $\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ Calculate this area, set the result equal to $\frac{P}{4}$, and solve the resulting algebraic equation.

Step 3: Detailed Explanation:
The given coordinates for the vertices are: Vertex 1: $(x_1, y_1) = (3, 8)$ Vertex 2: $(x_2, y_2) = (-4, 2)$ Vertex 3: $(x_3, y_3) = (5, 1)$ Let's substitute these values into the area formula: \[ \text{Area} = \frac{1}{2} |3(2 - 1) + (-4)(1 - 8) + 5(8 - 2)| \] Compute the differences inside the parentheses: \[ \text{Area} = \frac{1}{2} |3(1) - 4(-7) + 5(6)| \] Perform the multiplications: \[ \text{Area} = \frac{1}{2} |3 + 28 + 30| \] Sum the values inside the absolute value bars: \[ \text{Area} = \frac{1}{2} |61| \] Since 61 is positive, the absolute value is just 61: \[ \text{Area} = \frac{61}{2} \] The problem states that the area is equivalent to the expression $\frac{P}{4}$. We can set up an equation: \[ \frac{61}{2} = \frac{P}{4} \] To isolate and solve for $P$, multiply both sides of the equation by 4: \[ P = \frac{61}{2} \times 4 \] \[ P = 61 \times \left(\frac{4}{2}\right) \] \[ P = 61 \times 2 \] \[ P = 122 \]

Step 4: Final Answer:
The value of $P$ is $122$.