Question

The area of the triangle bounded by the lines $x = 4$, $y = -4$ and $y = x$ is

• A. 28
• B. 42
• C. 24
• D. 32
• E. 40

Hint:

When the triangle has a vertical or horizontal side, you can also use $\text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height}$. Here base $= 8$ (from $y=-4$ to $y=4$) and height $= 8$ (from $x=-4$ to $x=4$), giving $\dfrac{1}{2}\times 8\times 8 = 32$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Find the three vertices of the triangle formed by the intersections of the given lines, then use the standard area formula.

Step 2: Detailed Explanation:
Intersection 1: $x=4$ and $y=x$ $\Rightarrow$ $(4,\,4)$.
Intersection 2: $x=4$ and $y=-4$ $\Rightarrow$ $(4,\,-4)$.
Intersection 3: $y=x$ and $y=-4$ $\Rightarrow$ $(-4,\,-4)$.
Using the formula $\text{Area} = \dfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$:
\[ = \frac{1}{2}|4(-4-(-4)) + 4((-4)-4) + (-4)(4-(-4))| = \frac{1}{2}|0 + 4(-8) + (-4)(8)| = \frac{1}{2}|{-32-32}| = \frac{1}{2}\times 64 = 32 \]

Step 3: Final Answer:
Area of the triangle $= 32$ square units.