A piece of a broken wheel is shown. It was taken to a machine shop to be replaced with a new “whole” wheel. Find the radius of the wheel AC=10cm, BD=3 cm and D is the midpoint of AC. Round to the nearest hundredth.
To find the radius of the whole wheel, we can use the relationship between the radius and the lengths AC and BD.
Let's consider the triangle ABD. Since D is the midpoint of AC, we can infer that BD is the perpendicular bisector of AC. Therefore, the triangle ABD is a right triangle.
We can use the Pythagorean theorem to find the length of AB, which is equal to the radius of the wheel.
According to the Pythagorean theorem, the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AD and BD).
AB^2 = AD^2 + BD^2
First, let's find the length of AD. Since D is the midpoint of AC, AD is half the length of AC.
AD = AC/2 = 10cm/2 = 5cm
Now we can substitute the values into the Pythagorean theorem equation:
AB^2 = (5cm)^2 + (3cm)^2
AB^2 = 25cm^2 + 9cm^2
AB^2 = 34cm^2
To find AB, we take the square root of both sides of the equation:
AB = √34cm
Now, let's round the result to the nearest hundredth:
AB ≈ 5.83cm
Therefore, the radius of the whole wheel is approximately 5.83cm.