AB touches circle (P, 12) at A and circle (Q, 5) at B. If PQ= 25 and A&B are either side of PQ tthen AB= -------- please explain with diagram
To find the length of AB, you can use the concept of tangents and the properties of right triangles.
First, draw a diagram with two circles, (P) and (Q), and a line segment AB that touches both circles at points A and B.
Label the center of circle (P) as point P and the center of circle (Q) as point Q. The radius of circle (P) is given as 12, and the radius of circle (Q) is given as 5.
Now, draw a line segment PQ connecting the centers of the two circles. The length of PQ is given as 25.
Since A is on circle (P) and B is on circle (Q), both points lie on the circumference of their respective circles. This means that PA is a radius of circle (P), and QB is a radius of circle (Q).
As stated in the problem, points A and B are on opposite sides of the line segment PQ. So, AB will be the line segment that connects point A and point B.
To find the length of AB, we will need to use the properties of tangents and the right triangle formed by PA, AB, and QB.
The length of PA is equal to the radius of circle (P), which is given as 12. Similarly, the length of QB is equal to the radius of circle (Q), which is given as 5.
Now, we can apply the Pythagorean theorem in the right triangle formed by PA, AB, and QB.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, AB is the hypotenuse, so we can write:
AB^2 = PA^2 + QB^2
Substituting the given values, we get:
AB^2 = 12^2 + 5^2
AB^2 = 144 + 25
AB^2 = 169
Taking the square root of both sides, we find:
AB = √169
AB = 13
Therefore, the length of AB is 13 units.