Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

Four lines intersect, forming eight equal angles as shown in the diagram. Which black arc has the same length as the perimeter of the small gray circle?  So we know that perimeter and arc length are both one-dimensional quantities. So if we scale a figure by some amount, then the perimeter is going to get scaled by that same amount.  So going from the inner circle to the very outer circle, we scale the radius by four. That means our perimeter also gets scaled by a factor of four.  So the perimeter of this inner circle is one-quarter of the perimeter of the outer circle. Arc D is exactly one-quarter of the outer circle, so our final answer is D.

Video Summary

The problem involves four intersecting lines creating eight equal angles, with a circle diagram used in the solution. To find which black arc matches the perimeter of a smaller gray circle, we consider the scaling relationship in geometry. When a figure's dimensions are scaled by a factor, its perimeter scales by the same factor. Here, the radius from the inner to the outer circle is scaled by four, suggesting the inner circle's perimeter is one-quarter of the outer. Therefore, the arc that is equal to the inner circle's perimeter is Arc D, as it is one-quarter of the outer circle.

Keywords

intersecting lines

equal angles

scaling relationship

circle diagram

perimeter comparison