Question

Consider the following construction: (1) Begin with a square □ABCD with side length 1 . (2) Draw the midpoint of AB, call it M. (3) Draw the circle centered at M through C. (4) Draw AB, and mark the intersection of this ray with the circle as point E. Your problem: calculate the length of AE. You have seen this before.

Answer 1

To calculate the length of AE, we can use the concept of circumference and angles in a circle. In this construction, AC is the diameter of the circle, and AB is a chord that intersects the circle at point E. We can use the formula c = 2πr, where c is the circumference and r is the radius, to find the length of the arc AC. Since AB is a chord, it subtends an angle at the center of the circle equal to the angle at point C. This allows us to approximate the length of AB as the length of the corresponding arc AC. So, the length of AE, which is half of AB, can be approximated as half the length of AC.

Step-by-step explanation:

To calculate the length of AE, we can use the concept of circumference and angles in a circle. In this construction, AC is the diameter of the circle, and AB is a chord that intersects the circle at point E. We can use the formula c = 2πr, where c is the circumference and r is the radius, to find the length of the arc AC. Since AB is a chord, it subtends an angle at the center of the circle equal to the angle at point C. This allows us to approximate the length of AB as the length of the corresponding arc AC. So, the length of AE, which is half of AB, can be approximated as half the length of AC.