for a project, carlos must provide one cut-out paper model of each type of triangle: acute, right, and…

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.\nwhat is the total area of the three triangles?\n12.5 square units\n13 square units\n13.5 square units\n14 square units

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.\nwhat is the total area of the three triangles?\n12.5 square units\n13 square units\n13.5 square units\n14 square units

Answer

Explanation:

Step1: Identify top triangle vertices

Vertices: $(4,10)$, $(5,7)$, $(7,8)$ Use shoelace formula: $$\text{Area} = \frac{1}{2}\left|x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1)\right|$$ $$=\frac{1}{2}\left|47 + 58 + 710 - (105 + 77 + 84)\right|$$ $$=\frac{1}{2}\left|28 + 40 + 70 - (50 + 49 + 32)\right|=\frac{1}{2}\left|138 - 131\right|=\frac{7}{2}=3.5$$

Step2: Identify left bottom triangle vertices

Vertices: $(3,6)$, $(3,3)$, $(6,3)$ This is a right triangle: base $=3$, height $=3$ $$\text{Area}=\frac{1}{2}33=4.5$$

Step3: Identify right bottom triangle vertices

Vertices: $(7,6)$, $(7,1)$, $(9,4)$ Use shoelace formula: $$\text{Area}=\frac{1}{2}\left|71 + 74 + 96 - (67 + 19 + 47)\right|$$ $$=\frac{1}{2}\left|7 + 28 + 54 - (42 + 9 + 28)\right|=\frac{1}{2}\left|89 - 79\right|=5$$

Step4: Sum all triangle areas

$$\text{Total Area}=3.5 + 4.5 + 5$$

Answer:

13 square units