51. The figure below illustrates an isosceles right
triangle with legs of length 1, along with onefourth of a circle centered at the right-angle
vertex of the triangle. Using the result that
the shortest path between two points is a line
segment, explain why this figure shows that
2* (square root of ) is < than pi
To explain why the figure shows that 2*(square root of 2) is less than pi, let's analyze the paths along this figure.
First, consider the hypotenuse of the isosceles right triangle. By the Pythagorean theorem, we can find its length:
hypotenuse^2 = leg^2 + leg^2
hypotenuse^2 = 1^2 + 1^2 = 2
hypotenuse = square root of 2
Now, observe the circle segment. It is one-fourth of a complete circle with a radius equal to the length of the triangle's leg. The circumference of a circle can be calculated using the formula:
circumference = 2 * pi * radius
In this case, the radius is 1. So, the circumference of the circle segment is:
circumference = 2 * pi * 1/4 = pi/2
Consider a point A at one endpoint of the hypotenuse and a point B at the midpoint of the circle's arc:
Path 1: The shortest path between A and B is a straight line segment along the hypotenuse of length square root of 2.
Path 2: The path where you follow the circumference of the circle segment from A to B.
Now, let's compare the lengths of these two paths:
Path 1: square root of 2
Path 2: pi/2
Since the length of Path 1 (square root of 2) is less than the length of Path 2 (pi/2), we can conclude that:
square root of 2 < pi/2
Multiplying both sides of the inequality by 2 gives:
2 * (square root of 2) < pi
Therefore, this figure demonstrates that 2 * (square root of 2) is less than pi.