Question

what is the area of the original triangle? square units

Explanation:

Step1: Calculate the area of the rectangle

Assume the grid has unit - length squares. The rectangle that encloses the triangle (by completing the figure) has length \(l = 5\) units and width \(w=3\) units. The area of a rectangle \(A_{rect}=l\times w\), so \(A_{rect}=5\times3 = 15\) square units.

Step2: Calculate the area of the non - blue regions

There are three non - blue right - angled triangles.

• The first non - blue triangle has base \(b_1 = 3\) units and height \(h_1=1\) unit. The area of a right - angled triangle \(A=\frac{1}{2}bh\), so \(A_1=\frac{1}{2}\times3\times1=\frac{3}{2}\) square units.
• The second non - blue triangle has base \(b_2 = 5\) units and height \(h_2 = 2\) units. So \(A_2=\frac{1}{2}\times5\times2 = 5\) square units.
• The third non - blue triangle has base \(b_3=2\) units and height \(h_3 = 2\) units. So \(A_3=\frac{1}{2}\times2\times2=2\) square units.

The total area of the non - blue regions \(A_{non - blue}=A_1 + A_2+A_3=\frac{3}{2}+5 + 2=\frac{3 + 10+4}{2}=\frac{17}{2}=8.5\) square units.

Step3: Calculate the area of the blue triangle

The area of the blue triangle \(A_{triangle}=A_{rect}-A_{non - blue}\). So \(A_{triangle}=15-8.5 = 6.5\) square units.

Answer:

\(6.5\)