When a triangle has a horizontal or vertical side, choosing that side as the base simplifies finding the height, which will be the difference in the y-coordinates or x-coordinates, respectively.
Step 1: Understanding the Concept:
This problem requires calculating the areas of two different triangles defined by coordinates on a Cartesian plane. Step 2: Key Formula or Approach:
The area of a triangle is given by the formula: Area \( = \frac{1}{2} \times \text{base} \times \text{height}\). We need to identify a base and the corresponding perpendicular height for each triangle. Step 3: Detailed Explanation: For Column A: Area of triangular region OPQ
The vertices are O(0,0), P(2,2), and Q(2,0).
Let's choose the segment OQ as the base. O is at \(x=0\) and Q is at \(x=2\), so the length of the base OQ is \(2 - 0 = 2\).
The height corresponding to this base is the perpendicular distance from point P to the x-axis, which is the y-coordinate of P. The height is 2.
The triangle OPQ is a right-angled triangle with the right angle at Q.
\[ \text{Area}_{OPQ} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2 \]
The area of triangle OPQ is 2. For Column B: Area of triangular region ORS
The vertices are O(0,0), R(4,0), and S(4,-1).
Let's choose the segment OR as the base. O is at \(x=0\) and R is at \(x=4\), so the length of the base OR is \(4 - 0 = 4\).
The height corresponding to this base is the perpendicular distance from point S to the x-axis, which is the absolute value of the y-coordinate of S. The height is \(|-1| = 1\).
The triangle ORS is a right-angled triangle with the right angle at R.
\[ \text{Area}_{ORS} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 1 = 2 \]
The area of triangle ORS is 2. Step 4: Final Answer:
Comparing the two areas:
Column A = 2
Column B = 2
The two quantities are equal.
