Three circles touch each other externally. Their centres make a triangle with sides of 11cm, 15cm and 19cm. What are the radii of the three circles?
draw a diagram. The distance between centers is the sum of the radii, so if the circles have radii a,b,c we have
a+b=11
a+c=15
b+c=19
Now it's easy to figure out a,b,c.
To find the radii of the three circles, we can make use of the properties of triangles formed by the centers of the circles.
In the given problem, we have a triangle with sides of lengths 11cm, 15cm, and 19cm. Let's call this triangle ABC, where A, B, and C are the centers of the circles.
To find the radii, we can use a technique called "Tangents to Circles."
Step 1: Draw lines from points A, B, and C to the opposite side of the triangle (extended if necessary) such that they form the radii of the circles. Let's call these lines AD, BE, and CF, respectively.
Step 2: Now, we have three tangents drawn from points D, E, and F to the circles. These tangents are perpendicular to the radii and touch the circles at points of tangency.
Step 3: From the given problem, we know that each pair of circles touches externally. This means that the distance between the centers of any two circles will be equal to the sum of their radii.
Step 4: Using this information, we can calculate the lengths of AD, BE, and CF. Here, AD = radius of circle 1, BE = radius of circle 2, and CF = radius of circle 3.
Step 5: In triangle ABC, we can find the lengths of AD, BE, and CF using the Pythagorean theorem.
Let's calculate AD, the radius of circle 1:
The longest side of the triangle is 19, which corresponds to the side opposite angle C. Let's consider triangle ADC. Applying the Pythagorean theorem, we get:
AD^2 = AC^2 - CD^2
AC = 19
CD = AB + BD, where AB = 11 (side opposite angle C in triangle ABC) and BD = BE + EC (sum of radii of circle 2 and circle 3)
CD = 11 + (BE + CF)
Substituting the values, we have:
AD^2 = 19^2 - (11 + BE + CF)^2
Similarly, you can find the lengths of BE and CF.
After finding these lengths (AD, BE, CF), you will get the respective radii of the three circles.