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 wheel, we can use the concept of similar triangles.
Let's denote the center of the wheel as O. Since D is the midpoint of AC, we can see that triangle ADO is a right triangle with AD as the hypotenuse.
Using the Pythagorean theorem, we can find the length of AD:
AD^2 = AO^2 + OD^2
Since OD is half of AC, which is 10 cm, OD = 5 cm. We can substitute the values into the equation:
AD^2 = AO^2 + 5^2
Now, let's consider triangle BDO. It is also a right triangle, and we can apply the Pythagorean theorem to it as well:
BD^2 + OD^2 = BO^2
Substituting the known values:
3^2 + 5^2 = BO^2
Now, we have two equations:
AD^2 = AO^2 + 5^2
3^2 + 5^2 = BO^2
To find the radius of the wheel, we need to find the length of AO. Let's solve the first equation for AO:
AO^2 = AD^2 - 5^2
AO^2 = 10^2 - 5^2
AO^2 = 100 - 25
AO^2 = 75
Taking the square root of both sides, we find:
AO ≈ √75
To approximate this value to the nearest hundredth, we can use a calculator or a mathematical software tool. The approximate value is:
AO ≈ 8.6603
Therefore, the radius of the wheel, which is the length of AO, is approximately 8.6603 cm when rounded to the nearest hundredth.