In the rectangular coordinate system above, if point (a, b), shown, and the two points (4a, b) and (2a, 2b), not shown, were connected by straight lines, then the area of the resulting triangular region, in terms of a and b, would be
Show Hint
When finding the area of a triangle from coordinates, always look for two points that share an x- or y-coordinate. This gives you a horizontal or vertical side, which makes identifying the base and height much simpler than using the distance formula on all three sides.
Step 1: Understanding the Concept:
This problem asks for the area of a triangle given the coordinates of its three vertices. A straightforward approach is to use the formula Area = \(\frac{1}{2} \times \text{base} \times \text{height}\). Step 2: Key Formula or Approach:
Area of a triangle = \(\frac{1}{2} \times \text{base} \times \text{height}\). We can simplify the calculation by choosing a base that is either horizontal or vertical. Step 3: Detailed Explanation:
The three vertices of the triangle are:
Vertex 1: \(P_1 = (a, b)\)
Vertex 2: \(P_2 = (4a, b)\)
Vertex 3: \(P_3 = (2a, 2b)\) 1. Choose a base.
Notice that vertices \(P_1\) and \(P_2\) have the same y-coordinate (\(b\)). This means the line segment connecting them is horizontal. This is a convenient choice for the base of the triangle.
The length of the base is the distance between \(P_1\) and \(P_2\), which is the difference in their x-coordinates:
\[ \text{base} = |4a - a| = |3a| = 3a \]
(Since the point (a, b) is shown in the first quadrant, we know \(a>0\)). 2. Determine the height.
The height of the triangle is the perpendicular distance from the third vertex, \(P_3=(2a, 2b)\), to the line containing the base (the line \(y=b\)).
The height is the difference in the y-coordinates:
\[ \text{height} = |2b - b| = |b| = b \]
(Since \(b>0\)). 3. Calculate the area.
Now apply the area formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
\[ \text{Area} = \frac{1}{2} \times (3a) \times (b) \]
\[ \text{Area} = \frac{3ab}{2} \] Step 4: Final Answer:
The area of the resulting triangular region is \(\frac{3ab}{2}\).
