Samir is a short put champion and is planning
to participate in the next Olympic Games. He
starts to practice and completes 49 throws
from a center point in different directions.
When he started to measure all the distances,
he found that every throw landed in a spot 1-ft
away from the previous one. In addition, the
line connecting center point and the new spot
always was perpendicular to the line
connecting the new spot and the previous
spot. If his last throw distance was 1 feet, how
much distance was his very first throw?
To find the distance of Samir's very first throw, we need to analyze the pattern of the distances between each throw.
From the given information, we know that each throw is 1-ft away from the previous one, forming a series of points. We also know that the line connecting the center point (where Samir started) and each new spot is always perpendicular to the line connecting the new spot and the previous spot.
By visualizing this pattern, we can see that Samir's throws form a regular polygon, specifically a circle centered at the initial point. The distance between each consecutive point along the circumference of the circle is 1-ft.
To determine the distance of the very first throw, we need to find the radius of this circle. Since Samir's last throw distance was 1 foot, that means the radius of the circle is equal to the distance from the center point to the last throw spot.
Now, we know that a line connecting the center of a circle to a point on its circumference is always perpendicular to the circumference of the circle. This tells us that the line connecting the center point to the last throw spot is perpendicular to the line connecting the last throw spot to the penultimate throw spot.
Since all the throws land 1-ft away from each other and the last throw distance is 1 foot, it means that the line connecting the center point to the last throw spot is equal in length to the distance from the last throw spot to the penultimate throw spot.
By connecting these dots, we can form right-angled triangles, with one side being the radius of the circle, the other side being the distance between each throw spot, and the hypotenuse being the line connecting the center point to the last throw spot.
Using the Pythagorean Theorem, we can relate the sides of this right-angled triangle:
(radius)^2 + (distance between throw spots)^2 = (line connecting center point to last throw spot)^2
Let's denote the radius of the circle as r, the distance between the throw spots as d, and the line connecting the center point to the last throw spot as L.
We know that L = 1ft (since it's the last throw distance).
So, we have:
r^2 + d^2 = L^2
Substituting the known values:
r^2 + 1^2 = 1^2
Simplifying the equation:
r^2 + 1 = 1
r^2 = 1 - 1
r^2 = 0
Taking the square root of both sides:
r = √0
r = 0
From the calculations, we can see that the radius of the circle (and thus the distance of Samir's very first throw) is 0 feet.
Therefore, the distance of Samir's very first throw is 0 feet.