Question

use the diagram showing points p and q. determine which point is the remaining vertex of a triangle with an area of 4 square units. r(2,0) s(-2,-1) t(-1,0) u(2,-2)

Explanation:

Step1: Find the length of base PQ

The distance between $P(-2,1)$ and $Q(2,1)$ is calculated using the distance formula for two - points on a horizontal line (since $y$-coordinates are the same). The distance $d = |x_2 - x_1|$, so $PQ=|2-(-2)| = 4$.

Step2: Use the triangle - area formula

The area of a triangle is $A=\frac{1}{2}bh$. We know $A = 4$ and $b = PQ=4$. Substituting into the formula $4=\frac{1}{2}\times4\times h$.

Step3: Solve for the height h

From $4=\frac{1}{2}\times4\times h$, we can simplify the right - hand side to get $4 = 2h$. Dividing both sides by 2 gives $h = 2$.

Step4: Check the distance from the line $y = 1$ for each point

The line containing $P$ and $Q$ is $y = 1$.

• For point $R(2,0)$: The distance from the line $y = 1$ is $|1 - 0|=1$.
• For point $S(-2,-1)$: The distance from the line $y = 1$ is $|1-(-1)| = 2$.
• For point $T(-1,0)$: The distance from the line $y = 1$ is $|1 - 0|=1$.
• For point $U(2,-2)$: The distance from the line $y = 1$ is $|1-(-2)| = 3$.

Answer:

B. $S(-2,-1)$