Let's work through the problems shown in your image step by step.

Problem 2:

Find the area of the triangles (a) and (b) and explain the reasoning.

• Formula for the area of a triangle:
  $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$ Here, the base and height are determined by counting the number of grid squares along the base and height of each triangle.

Triangle (a):

• Base: The base is along the horizontal axis, and counting the squares gives a base of 6 units.
• Height: The height is the vertical distance, which is 5 units.
• Using the area formula: $\text{Area of } a = \frac{1}{2} \times 6 \times 5 = 15 \, \text{square units}$

Triangle (b):

• Base: The base is 7 units (counting along the horizontal axis).
• Height: The height is 3 units (vertical distance).
• Using the area formula: $\text{Area of } b = \frac{1}{2} \times 7 \times 3 = 10.5 \, \text{square units}$

Problem 3:

Which of the three triangles (A, B, C) has the greatest area? Show your reasoning.

We can use the same method as above to find the areas of triangles A, B, and C by counting the base and height on the grid.

Triangle A:

• Base: 6 units
• Height: 5 units
• Using the area formula: $\text{Area of A} = \frac{1}{2} \times 6 \times 5 = 15 \, \text{square units}$

Triangle B:

• Base: 6 units
• Height: 4 units
• Using the area formula: $\text{Area of B} = \frac{1}{2} \times 6 \times 4 = 12 \, \text{square units}$

Triangle C:

• Base: 8 units
• Height: 3 units
• Using the area formula: $\text{Area of C} = \frac{1}{2} \times 8 \times 3 = 12 \, \text{square units}$

Conclusion for Problem 3:

• Triangle A has the greatest area (15 square units).

Would you like a more detailed explanation or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How does changing the base or height affect the area of a triangle?
2. Can a triangle with a smaller base than another have a larger area? Why?
3. What would happen to the area if both the base and height were doubled?
4. How do the areas of right triangles differ from other triangles?
5. Could you apply this method to find areas of shapes other than triangles?

Tip: When using grid-based problems, always count the grid units carefully to avoid mistakes in calculating base and height.