Question

Consider these two geoboard quadrilaterals. Which of the following statements is true?

The area of quadrilateral $I$ is more than the area of quadrilateral $II.$

The area of quadrilateral $I$ is less than the area of quadrilateral $II.$

The quadrilaterals have the same area and the same perimeter.

The quadrilaterals have the same area, but the perimeter of $I$ is more than the perimeter of $II.$

The quadrilaterals have the same area, but the perimeter of $I$ is less than the perimeter of $II.$

Solution:

Assume that the pegs on this grid are separated by $1$ unit.

Note that region $I$ is a parallelogram with base $1$ and height $1,$ makings its area $1 \cdot 1 = 1.$

We can split region $II$ into $2$ triangles. Both with base $1$ and height $1.$ This makes the sum of the areas $2 \cdot \dfrac{1}{2} \cdot 1 \cdot 1 = 1.$ This shows that both regions have the same area.

Note that each region has $2$ sides that are of length $\sqrt{2}.$ Region $I$ has $2$ unit sides, whereas region $II$ only has $1.$

The other side of region $II$ is clearly greater than $1,$ which shows that region $II$ has the greater perimeter.

Thus, E is the correct answer.

2000 AMC 8 Problem 18

Solution:

Problem 18 in Other Years