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

Explanation:

Step1: Calculate the area of the first triangle (right - triangle at the bottom)

The formula for the area of a triangle is \(A=\frac{1}{2}bh\). For the right - triangle with \(b = 4\) (from \(x = 3\) to \(x = 7\)) and \(h=3\) (from \(y = 3\) to \(y = 6\)), \(A_1=\frac{1}{2}\times4\times3 = 6\) square units.

Step2: Calculate the area of the second triangle (right - triangle on the left)

For the right - triangle with \(b = 3\) (from \(x = 6\) to \(x = 9\)) and \(h = 3\) (from \(y = 1\) to \(y = 4\)), \(A_2=\frac{1}{2}\times3\times3=4.5\) square units.

Step3: Calculate the area of the third triangle (the top - most triangle)

Use the formula \(A=\frac{1}{2}\times base\times height\). For the triangle with \(b = 4\) (horizontal distance) and \(h = 2\) (vertical distance), \(A_3=\frac{1}{2}\times3\times2 = 3\) square units.

Step4: Sum up the areas of the three triangles

\(A = A_1+A_2+A_3\). Substitute \(A_1 = 6\), \(A_2=4.5\) and \(A_3 = 3\) into the formula: \(A=6 + 4.5+3=13.5\) square units.

Answer:

13.5 square units