Question

for a project, carlos must provide one cut-out paper model of each type of triangle: acute, right, and obtuse. he sketches each triangle on graph paper before making the models.

what is the total area of the three triangles?

12.5 square units
13 square units
13.5 square units
14 square units

Answer:

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<reasoning>

Find coordinates of the first triangle

Using the Triangle Area in Coordinate Plane knowledge point
\[

$$\begin{aligned} &\text{Vertices: } (3, 3), (6, 3), (3, 6)\\ &\text{Base } b = 6 - 3 = 3\\ &\text{Height } h = 6 - 3 = 3\\ &\text{Area}_1 = \frac{1}{2} \times 3 \times 3 = 4.5 \end{aligned}$$

\]

Find coordinates of the second triangle

Using the Triangle Area in Coordinate Plane knowledge point
\[

$$\begin{aligned} &\text{Vertices: } (7, 1), (7, 6), (9, 4)\\ &\text{Base } b = 6 - 1 = 5\\ &\text{Height } h = 9 - 7 = 2\\ &\text{Area}_2 = \frac{1}{2} \times 5 \times 2 = 5 \end{aligned}$$

\]

Find coordinates of the third triangle

Using the Triangle Area in Coordinate Plane knowledge point
\[

$$\begin{aligned} &\text{Vertices: } (3.5, 10), (4.8, 7), (7, 8)\\ &\text{Bounding Box: } x \in [3.5, 7], y \in [7, 10]\\ &\text{Box Area} = (7 - 3.5) \times (10 - 7) = 3.5 \times 3 = 10.5\\ &\text{Subtract outer right triangles:}\\ &T_1 \text{ (top-left to bottom): } \frac{1}{2} \times (4.8 - 3.5) \times (10 - 7) = \frac{1}{2} \times 1.3 \times 3 = 1.95\\ &T_2 \text{ (bottom to right): } \frac{1}{2} \times (7 - 4.8) \times (8 - 7) = \frac{1}{2} \times 2.2 \times 1 = 1.1\\ &T_3 \text{ (top-left to right): } \frac{1}{2} \times (7 - 3.5) \times (10 - 8) = \frac{1}{2} \times 3.5 \times 2 = 3.5\\ &\text{Area}_3 = 10.5 - (1.95 + 1.1 + 3.5) = 10.5 - 6.55 = 3.95 \approx 4 \end{aligned}$$

\]
Let's re-examine the grid coordinates for the third triangle carefully:

• Top-left vertex: \(x = 3.5\) or \(x = 3.6\)? Looking closely at the grid, the top-left vertex is at \(x = 3.6\), \(y = 10\).
• Bottom vertex: \(x = 4.8\) or \(x = 4.8\), \(y = 7\).
• Right vertex: \(x = 7\), \(y = 8\).

Let's use the Shoelace Formula with exact grid values:
Top-left: \((3.6, 10)\)
Bottom: \((4.8, 7)\)
Right: \((7, 8)\)
Using the Shoelace Formula knowledge point
\[

$$\begin{aligned} \text{Area}_3 &= \frac{1}{2} |3.6(7 - 8) + 4.8(8 - 10) + 7(10 - 7)|\\ &= \frac{1}{2} |-3.6 - 9.6 + 21|\\ &= \frac{1}{2} |7.8| = 3.9 \end{aligned}$$

\]
Let's check if the top-left vertex is at \((3.6, 10)\) or \((3.5, 10)\).
If \((3.6, 10)\), \((4.8, 7)\), \((7, 8)\): Area is \(3.9\).
If \((3.5, 10)\), \((4.8, 7)\), \((7, 8)\): Area is \(3.95\).
If \((3.6, 10)\), \((4.8, 7)\), \((7, 8)\) are slightly off, let's look at the options:

• \(12.5\) square units
• \(13\) square units
• \(13.5\) square units
• \(14\) square units

Let's sum the areas:
\[
\text{Total Area} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3
\]
If \(\text{Area}_3 = 4\):
\[
\text{Total Area} = 4.5 + 5 + 4 = 13.5
\]
If \(\text{Area}_3 = 3.9\):
\[
\text{Total Area} = 4.5 + 5 + 3.9 = 13.4 \approx 13.5
\]
Thus, the total area is \(13.5\) square units.

Sum the composite areas

Using the Composite Area knowledge point
\[

$$\begin{aligned} \text{Total Area} &= 4.5 + 5.0 + 4.0\\ &= 13.5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>12.5 square units</mcq-option>
<mcq-option>13 square units</mcq-option>
<mcq-correct>13.5 square units</mcq-correct>
<mcq-option>14 square units</mcq-option>
</answer>

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