Question

triangle ijk, with vertices i(-8,6), j(-4,9), and k(-9,8), is drawn inside a rectangle, as shown below.
what is the area, in square units, of triangle ijk?

Explanation:

Step1: Find rectangle dimensions

Vertices of rectangle: leftmost x=-9, rightmost x=-4, bottom y=6, top y=9. Width = $|-4 - (-9)| = 5$, height = $9 - 6 = 3$. Area = $5×3 = 15$.

Step2: Calculate area of 3 right triangles

Triangle 1 (I, bottom-left, K): base=1, height=2. Area=$\frac{1×2}{2}=1$.
Triangle 2 (K, top-left, J): base=5, height=1. Area=$\frac{5×1}{2}=2.5$.
Triangle 3 (I, bottom-left, J): base=5, height=3. Area=$\frac{5×3}{2}=7.5$.
Sum of triangle areas: $1 + 2.5 + 7.5 = 11$.

Step3: Compute triangle IJK area

Area = Rectangle area - Sum of 3 triangles: $15 - 11 = 4$? Correction: Use shoelace formula for accuracy:
Vertices I(-8,6), J(-4,9), K(-9,8):
Area = $\frac{1}{2}|(-8×9 + (-4)×8 + (-9)×6) - (6×(-4) + 9×(-9) + 8×(-8))|$
= $\frac{1}{2}|(-72 -32 -54) - (-24 -81 -64)|$
= $\frac{1}{2}|(-158) - (-169)| = \frac{1}{2}|11| = 5.5$? Correction: Correct shoelace:
$\frac{1}{2}|x_I(y_J - y_K) + x_J(y_K - y_I) + x_K(y_I - y_J)|$
= $\frac{1}{2}|(-8)(9-8) + (-4)(8-6) + (-9)(6-9)|$
= $\frac{1}{2}|(-8)(1) + (-4)(2) + (-9)(-3)|$
= $\frac{1}{2}|-8 -8 +27| = \frac{1}{2}|11| = 5.5$? Wait, correct: $-8(1)=-8$, $-4(2)=-8$, $-9(-3)=27$. Sum: $-8-8+27=11$. Half is 5.5. Final correct answer is 5.5? Wait, original rectangle method: Correct rectangle vertices from graph: K(-9,8), J(-4,9), bottom-left (-9,6), bottom-right (-4,6). So rectangle area: width=5, height=3 →15. Triangles:

1. I(-8,6) to (-4,6) to J(-4,9): area $\frac{1}{2}×4×3=6$.
2. I(-8,6) to (-9,6) to K(-9,8): area $\frac{1}{2}×1×2=1$.
3. K(-9,8) to (-9,9) to J(-4,9): area $\frac{1}{2}×5×1=2.5$.

Sum:6+1+2.5=9.5. 15-9.5=5.5. Yes, so final answer 5.5.

Step1: Use shoelace formula

Formula: $\frac{1}{2}|x_I(y_J - y_K) + x_J(y_K - y_I) + x_K(y_I - y_J)|$

Step2: Substitute vertices

$x_I=-8,y_I=6;x_J=-4,y_J=9;x_K=-9,y_K=8$
$\frac{1}{2}|(-8)(9-8)+(-4)(8-6)+(-9)(6-9)|$

Step3: Calculate

$\frac{1}{2}|-8 -8 +27|=\frac{1}{2}×11=5.5$

Answer:

3.5