Question

1. find the area of the shape below. justify your answer.

Explanation:

To solve the problem of finding the area of the blue triangle, we can use the grid method (or the "subtracting areas of surrounding shapes from the rectangle" method). Let's assume each grid square has a side length of 1 unit.

Step 1: Determine the dimensions of the rectangle \( RMNP \)

Looking at the grid, the length of \( RMNP \) (horizontal) is 10 units, and the height (vertical) is 6 units.
Area of rectangle \( RMNP = \text{length} \times \text{height} = 10 \times 6 = 60 \) square units.

Step 2: Calculate the area of the three surrounding right triangles

We subtract the areas of the three non - blue right triangles from the rectangle’s area to find the area of the blue triangle.

Triangle 1 (top - left, with vertex \( R, Q, M \)):

Base \( = 4 \) units, Height \( = 6 \) units.
Area of a triangle \( = \frac{1}{2} \times \text{base} \times \text{height} \)
So, area \( = \frac{1}{2} \times 4 \times 6 = 12 \) square units.

Triangle 2 (top - right, with vertex \( Q, P, O \)):

Base \( = 6 \) units, Height \( = 2 \) units.
Area \( = \frac{1}{2} \times 6 \times 2 = 6 \) square units.

Triangle 3 (bottom - right, with vertex \( O, N, M \)):

Base \( = 10 \) units, Height \( = 4 \) units.
Area \( = \frac{1}{2} \times 10 \times 4 = 20 \) square units.

Step 3: Subtract the areas of the three triangles from the rectangle’s area

Area of blue triangle \( = \text{Area of rectangle} - (\text{Area of Triangle 1} + \text{Area of Triangle 2} + \text{Area of Triangle 3}) \)
\( = 60-(12 + 6+20) \)
\( = 60 - 38 \)
\( = 22 \) square units.

Answer:

The area of the blue shape (triangle) is \(\boldsymbol{22}\) square units (assuming each grid square has side length 1 unit).