Question

The area of the triangle whose vertices are \((3,8)\), \((-4,2)\) and \((5,1)\) is

• A. \(63/2\)
• B. \(61/2\)
• C. \(59/2\)
• D. \(57/2\)

Hint:

For triangle area from coordinates, use: \[ \frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right| \]

Answer 1

The Correct Option is B

Concept:
The area of a triangle with vertices: \[ (x_1,y_1),\quad (x_2,y_2),\quad (x_3,y_3) \] is given by: \[ \text{Area}=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right| \]

Step 1: Write the given points.
The vertices are: \[ (3,8),\quad (-4,2),\quad (5,1) \] So: \[ x_1=3,\quad y_1=8 \] \[ x_2=-4,\quad y_2=2 \] \[ x_3=5,\quad y_3=1 \]

Step 2: Apply the area formula.
\[ \text{Area}=\frac12\left|3(2-1)+(-4)(1-8)+5(8-2)\right| \]

Step 3: Simplify each term.
First term: \[ 3(2-1)=3(1)=3 \] Second term: \[ (-4)(1-8)=(-4)(-7)=28 \] Third term: \[ 5(8-2)=5(6)=30 \]

Step 4: Add the terms.
\[ 3+28+30=61 \] So: \[ \text{Area}=\frac12|61| \] \[ \text{Area}=\frac{61}{2} \]

Step 5: Check the options.
Option (A) \(63/2\) is incorrect.
Option (B) \(61/2\) is correct.
Option (C) \(59/2\) is incorrect.
Option (D) \(57/2\) is incorrect. Hence, the correct answer is: \[ \boxed{(B)\ \frac{61}{2}} \]