Two circles with centers a and b of radii 3 cm and 4 cm res , intersecting at two points c and d such that ac and bc are two tangets of the circles, then length of common chord .
how can a line though the center (ac) be a tangent to the circle?
IT IS TANGENT TO OTHER CIRCLE.
Hmmm. OK. In that case, the two circles must be orthogonal to each other. That is, if you draw triangle acb and adb, each is a 3-4-5 right triangle. The common chord is just twice the altitude to the hypotenuse, which I'm sure you can find.
To find the length of the common chord, we can use the properties of tangents drawn to a circle.
First, let's draw a diagram to visualize the given information:
```
a
O------O
| |
| |
c O------O d
| |
| |
O------O
b
```
Here, O represents the centers of the circles, a and b. The radii of the circles are given as 3 cm and 4 cm. The tangents from point a and b are represented by ac and bc, respectively. The points where the circles intersect are labeled as c and d.
To find the length of the common chord, we need to use the following property:
In a circle, if two tangents are drawn from an external point, then the lengths of the tangent segments are equal.
In our case, the tangents ac and bc are drawn from the external point a and b, respectively. Therefore, the lengths of ac and bc will be equal.
To find the length of the common chord, we can consider triangle acd. Here, ac and bc are equal since they are tangents to the circles.
Let's use Pythagoras' Theorem to find the length of ac:
ac² = (ad)² + (cd)²
Since ac = bc, we can rewrite the equation as:
(bc)² = (ad)² + (cd)²
Since ad and cd form a diameter, their sum is equal to the diameter of the circle with radius 4 cm:
(ad) + (cd) = 2 * 4 = 8
Now we can substitute this value in the equation:
(bc)² = 8² - (cd)²
Simplifying further:
(bc)² = 64 - (cd)²
Since bc = ac:
(ac)² = 64 - (cd)²
Since ac = 3 cm:
3² = 64 - (cd)²
9 = 64 - (cd)²
Rearranging the equation:
(cd)² = 64 - 9
(cd)² = 55
Taking the square root of both sides:
cd = √55
Thus, the length of the common chord is √55 cm.