We have two circles that are touching each other from the outside. Let’s find out how far their touching point is from the special lines that touch both circles!
Problem:
Two circles with radii R=3 cm and r=1 cm touch externally. Find the distance from their point of tangency to their common tangents.
Solution:
Case 1: The Common Internal Tangent
Imagine our two circles are like two friends holding hands. The point where their hands touch is their “point of tangency” (M). Now, if you draw a line that touches both circles right at this point M, this is called the common internal tangent. Since this line passes directly through their touching point, the distance from the touching point to this line is simply 0 cm.
Case 2: The Common External Tangents
Now, let’s think about the lines that touch both circles from the outside, like a rope tied around them. These are called the common external tangents. There are two such lines, but their distance from the point of tangency will be the same.
Let’s use a clever trick from geometry to find this distance! We’ll use a coordinate system, which is like drawing an invisible graph to help us measure things.
1. Set up our coordinate system: Let’s place one of the common external tangents right on the X-axis of our graph. This means the equation of this tangent line is y=0.
2. Locate the centers of the circles:
- The center of the larger circle, O_1, will have a y-coordinate equal to its radius, R. So, O_1 = (x_1, R).
- The center of the smaller circle, O, will have a y-coordinate equal to its radius, r. So, O = (x_2, r).
3. Understand the relationship between the centers:
- The distance between the centers, O_1O, is simply the sum of their radii because they touch externally: O_1O = R+r.
- We also know that the horizontal distance between the points where the circles touch the X-axis (K for the larger circle and Q for the smaller) is given by the formula for the length of the common external tangent segment: KQ = 2\sqrt{Rr}. So, we can set x_1=0 and x_2=2\sqrt{Rr}.
- Thus, our centers are O_1 = (0, R) and O = (2\sqrt{Rr}, r).
4. Find the point of tangency M: The point where the two circles touch, M, lies exactly on the straight line connecting their centers, O_1O. This point M divides the segment O_1O in a special ratio: R:r. This means O_1M = R and MO = r.
5. Calculate the y-coordinate of M: The y-coordinate of M will give us the perpendicular distance from M to our X-axis (which is our common tangent!). We use a formula called the “section formula” for this:
y_M = \frac{r \cdot y_{O_1} + R \cdot y_O}{r+R}Let’s plug in the y-coordinates of our centers:
y_M = \frac{r \cdot R + R \cdot r}{r+R} y_M = \frac{2Rr}{R+r}This is the general formula for the distance from the point of tangency to a common external tangent.
6. Substitute the given values:
We are given R=3 cm and r=1 cm.
h = y_M = \frac{2 \cdot 3 \cdot 1}{3+1} h = \frac{6}{4}h = \frac{3}{2} cm
Answer:
The distance from the point of tangency of the circles to their common internal tangent is 0 cm. The distance from the point of tangency of the circles to their common external tangents is \frac{3}{2} cm.
