Question

If A(0, 1), B(0, 5) and C(3, 4) are the vertices of any \(\triangle ABC\), then the area (in square unit) of \(\triangle ABC\) is

• A. 16
• B. 12
• C. 6
• D. 4

Hint:

Before jumping into the full area formula, always check if two of the vertices share the same x or y coordinate. If so, you have a vertical or horizontal base, and the `1/2 * base * height` method is much faster.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the area of a triangle given the coordinates of its three vertices. We can use the standard coordinate geometry formula for the area of a triangle, or a simpler method if the triangle has a vertical or horizontal base.

Step 2: Key Formula or Approach:
Method 1: Area Formula
\[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \] Method 2: Base and Height
Notice that points A(0, 1) and B(0, 5) both have an x-coordinate of 0. This means they lie on the y-axis, and the segment AB is a vertical line. We can use this as the base of the triangle.
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

Step 3: Detailed Explanation:
Using Method 2 (Base and Height), which is simpler here:
The base of the triangle is the distance between A(0, 1) and B(0, 5).
\[ \text{Base} = |5 - 1| = 4 \text{ units} \] The height of the triangle is the perpendicular distance from the third vertex C(3, 4) to the line containing the base (the y-axis). This distance is simply the absolute value of the x-coordinate of C.
\[ \text{Height} = |3| = 3 \text{ units} \] Now, calculate the area:
\[ \text{Area} = \frac{1}{2} \times 4 \times 3 = 6 \text{ square units} \]

Step 4: Final Answer:
The area of \(\triangle ABC\) is 6 square units.