Q2(b):

Find the area of each of the following triangles:

(b)     

Given Variables & Initial Setup

Based on the principles of Coordinate Geometry, when a geometric figure is presented on a Cartesian plane, its dimensions must be extracted by identifying the coordinates of its vertices. For this analysis, we extract the coordinates of the triangle's vertices from the provided two-dimensional grid.

• Vertex $A$ (Apex): $(0, 5)$
• Vertex $B$ (Left base endpoint): $(-4, 0)$
• Vertex $C$ (Right base endpoint): $(3, 0)$

[Theoretical Justification: In a Cartesian system, the distance between two points lying on the same horizontal or vertical axis can be determined using the absolute difference of their respective non-zero coordinates, per the 1D Distance Formula $d = |a_2 - a_1|$].

Step 1: Determining the Length of the Base ($b$)

The base of the triangle, segment $BC$, lies entirely on the x-axis. Because both points share the same y-coordinate ($y = 0$), the length of the base is the absolute difference between their x-coordinates.

Let $x_B = -4$ and $x_C = 3$.

$ \text{Base } (b) = |x_C - x_B| $

$ b = |3 - (-4)| $

$ b = |3 + 4| = 7 \text{ units} $

Step 2: Determining the Altitude / Height ($h$)

The height of a triangle is the perpendicular distance from the apex to the line containing the base. Since the base lies on the x-axis, the perpendicular distance from vertex $A(0, 5)$ to the x-axis is simply the absolute value of its y-coordinate.

Let $y_A = 5$ and the y-coordinate of the base $y_{base} = 0$.

$ \text{Height } (h) = |y_A - y_{base}| $

$ h = |5 - 0| = 5 \text{ units} $

Step 3: Calculating the Area of the Triangle

According to Euclidean geometry, the area ($A$) of a triangle is given by half the product of its base and its corresponding altitude.

$ A = \frac{1}{2} \times b \times h $

Substituting the derived values:

$ A = \frac{1}{2} \times 7 \times 5 $

$ A = \frac{1}{2} \times 35 $

$ A = 17.5 \text{ square units} $

Step 4: Rigorous Verification via Coordinate Geometry Formula

To ensure absolute precision, we verify the result using the determinant-based area formula for a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$:

$ A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| $

Assigning the coordinates: $(x_1, y_1) = (0, 5)$, $(x_2, y_2) = (-4, 0)$, $(x_3, y_3) = (3, 0)$.

$ A = \frac{1}{2} |0(0 - 0) + (-4)(0 - 5) + 3(5 - 0)| $

$ A = \frac{1}{2} |0 + (-4)(-5) + 3(5)| $

$ A = \frac{1}{2} |0 + 20 + 15| $

$ A = \frac{1}{2} |35| = 17.5 \text{ square units} $

[The identical result confirms the geometric extraction is mathematically sound and verified across multiple theorems].

Final Solution: The area of the given triangle is 17.5 square units.