Question

The area of the triangular region whose sides are \( y = 2x + 1 \), \( y = 3x + 1 \) and \( x = 4 \) is:

• A. \( 5 \)
• B. \( 6 \)
• C. \( 7 \)
• D. \( 8 \)
• E. \( 9 \)

Hint:

When one side of a triangle is perfectly vertical or horizontal, identifying that side as the base significantly simplifies the area calculation, avoiding the complex determinant formula.

Answer 1

The Correct Option is D

Concept: The area of a triangle formed by three lines can be found by identifying the coordinates of the three vertices where the lines intersect. Once the vertices are known, we can treat the vertical line as the base and use the horizontal distance to the third vertex as the height.

Step 1: Find the vertices of the triangle.

• Intersection of \( y = 2x + 1 \) and \( y = 3x + 1 \): Set \( 2x + 1 = 3x + 1 \implies x = 0 \). Substituting \( x=0 \) into either equation gives \( y = 1 \). Vertex 1: \( (0, 1) \).
• Intersection of \( y = 2x + 1 \) and \( x = 4 \): Substitute \( x = 4 \) into the equation: \( y = 2(4) + 1 = 9 \). Vertex 2: \( (4, 9) \).
• Intersection of \( y = 3x + 1 \) and \( x = 4 \): Substitute \( x = 4 \) into the equation: \( y = 3(4) + 1 = 13 \). Vertex 3: \( (4, 13) \).

Step 2: Calculate the area using the base and height.
The triangle has two vertices on the vertical line \( x = 4 \). We can treat the segment between these two points as the base: \[ \text{Base} = |13 - 9| = 4 \] The height of the triangle is the horizontal distance from the third vertex \( (0, 1) \) to the line \( x = 4 \): \[ \text{Height} = |4 - 0| = 4 \] Using the area formula \( \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \): \[ \text{Area} = \frac{1}{2} \times 4 \times 4 = 8 \]