Question

1. which triangle has the greatest area? the least area? explain your reasoning.

Explanation:

Step1: Recall triangle area formula

The area formula of a triangle is $A=\frac{1}{2}bh$, where $b$ is the base and $h$ is the height.

Step2: Assume grid - unit length

Let the length of each grid - square side be 1 unit.

Step3: Calculate area of triangle A

For triangle A, assume the base $b = 2$ units and the height $h=3$ units. Then $A_A=\frac{1}{2}\times2\times3 = 3$ square units.

Step4: Calculate area of triangle B

For triangle B, assume the base $b = 3$ units and the height $h = 3$ units. Then $A_B=\frac{1}{2}\times3\times3=\frac{9}{2}=4.5$ square units.

Step5: Calculate area of triangle C

For triangle C, assume the base $b = 4$ units and the height $h = 3$ units. Then $A_C=\frac{1}{2}\times4\times3=6$ square units.

Step6: Calculate area of triangle D

For triangle D, assume the base $b = 3$ units and the height $h = 3$ units. Then $A_D=\frac{1}{2}\times3\times3 = 4.5$ square units.

Step7: Compare the areas

We have $A_A = 3$, $A_B=4.5$, $A_C = 6$, $A_D=4.5$. Since $6>4.5>3$, triangle C has the greatest area and triangle A has the least area.

Answer:

Triangle C has the greatest area and triangle A has the least area. The reason is that using the formula $A=\frac{1}{2}bh$ for the area of a triangle and calculating the areas of each triangle based on the grid - unit lengths of their bases and heights, we find that the area of triangle C is 6 square units, the areas of triangles B and D are 4.5 square units each, and the area of triangle A is 3 square units.