Questions: Find the area of the triangle defined by the points (2,1)(5,1)(2,8).

Solution Steps

To find the area of a triangle given its vertices, we can use the formula for the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

This formula calculates the absolute value of the determinant of a matrix formed by the coordinates, divided by 2.

The vertices of the triangle are given as \( (2, 1) \), \( (5, 1) \), and \( (2, 8) \).

Using the area formula for a triangle defined by vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x3(y_1-y_2) \right| \]

Substituting the coordinates:

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

Calculating the expression inside the absolute value:

\[ = \frac{1}{2} \left| 2(-7) + 5(7) + 2(0) \right| = \frac{1}{2} \left| -14 + 35 + 0 \right| = \frac{1}{2} \left| 21 \right| = \frac{21}{2} = 10.5 \]

Final Answer