Draw a line from the point where the third baseman stands to first base to form a right triangle. Then, use the Pythagorean Theorem to find the hypotenuse.
C
c
Draw a line from home plate to the point where the base runner stands to form a right triangle. Then, use the Pythagorean Theorem to find the measure of the missing leg.
A
a
≈ 127 feet
B
b
≈ 117 feet
C
c
≈ 46 feet
Practice makes perfect
a
We are given a baseball diamond in the shape of a square. Let's first consider this field.
In this baseball diamond, a catcher wants to throw the ball from home plate to second base. We want to find out how far the ball has to travel. To do so, let's examine the right triangle that is formed on the right-hand side of the square baseball diamond.
Note that the path of the ball from home plate to the second base is the hypotenuse of the right triangle. Therefore, we will use the Pythagorean Theorem to find the length of the hypotenuse.
a^2+b^2=c^2
In this form, a and b are the legs and c is the hypotenuse of a right triangle. Let's substitute a= 90 and b= 90 into the above formula and then isolate the hypotenuse c.
Since a negative side length does not make sense, we only need to consider positive solutions. Therefore, the catcher has to throw the ball about 127 feet away.
b
We are told that a third baseman throws the ball to the first baseman from a point on the baseline 15 feet from third to second base. This means that the third baseman is 75 feet away from second base. Since we have a square baseline, we can form a right triangle by drawing a line between the third and the first basemen.
We have the lengths of the two legs and we want to find the hypotenuse of the right triangle. Therefore, we will use the Pythagorean Theorem again.
a^2+b^2=c^2
Now, let's substitute a= 75 and b= 90 into the above formula and then isolate the hypotenuse c.
Since a negative side length does not make sense, we only need to consider positive solutions. Therefore, the third baseman needs to throw the ball about 117 feet to the first baseman.
c
In this case, the base runner going from first to second base is 100 feet from home plate. We want to find the distance between the second base and base runner. To do so, let's first draw a line from home plate to the base runner to compute the distance between the base runner and first base.
Notice that we have a right triangle on the right hand side of the figure. Now, we will use Pythagorean Theorem one more time to find the the missing leg, which is the distance between the base runner and first base.
a^2+b^2=c^2
Let's substitute a= 90 and c=100 into the above formula and then isolate the length of the missing leg b.
Since a negative side length does not make sense, we only need to consider positive solutions. Therefore, the distance between the base runner and first base is about 44 feet. Since the total distance between first and second base is 90 feet, we need to subtract 44 feet from 90 feet.
90ft-(≈ 44ft)≈ 46ft
The distance between the baseman and second base is about 46 feet.
