Circle C has radius 4 inches. Circle D had radius 9 inches. These two circles are connected. Point A,B,and E are points of tangency. Find AB. From point C to point D(which is the radius of circle C plus the raduis of circle D) it is 13 inches. From point C to point A (radius of circle C) it is 4 inches. From point D to point B it is 9 inches.(radius of circle D) What is the length from point A to point B? (if you connect the bottom of the two circles which point A is the point of tangency in circle C from the radius down forming a 90 degree angle and also point B is the point of tangency in circle D from the radius down forming a 90 degree angle)
Thanks so much!
Where does point E enter the picture?
It is supposed to be a point of tangency, is it the point where the two circles are "connected" ?
It does not appear to enter into the calculation of AB.
From C draw a line parallel to AB to cut BD at F.
So FC = AB since we now have a rectangle where
angle DFC = 90°
by Pythagoras,
FC^2 + 5^2 = 13^2
FC = 12
so AB = 12
To find the length from point A to point B, you can use the concept of similar triangles. Here's how you can solve it step by step:
1. Draw a diagram of the circles and label the given points and lengths.
2. Recognize that triangle CAB and triangle DBE are similar since they share a right angle at point C (90 degrees) and they share the same angle at B and E (equal to the angle at C in triangle CAB).
3. Use the properties of similar triangles to set up a proportion. Let's represent the length AB as x:
AB / (AC + DB) = CA / CB (by the similarity of triangles CAB and DBE)
x / (4 + 9) = 4 / 9 (substituting the given lengths)
4. Solve the proportion for x:
x / 13 = 4 / 9
9x = 4 * 13
9x = 52
x = 52 / 9
Therefore, the length from point A to point B (AB) is 52/9 inches.