Two circles touch externally. The sum of their areas is 89π square cm and the distance between their centres is 13 cm. What is the difference in their radii?
Solving Circle Geometry Problems: Radii Difference
This problem involves two circles that are touching each other externally. We are given the sum of their areas and the distance between their centers, and we need to find the difference between their radii.
Understanding Externally Touching Circles
When two circles touch externally, the distance between their centers is equal to the sum of their radii. Let's denote the radii of the two circles as \(r_1\) and \(r_2\).
According to the problem, the distance between their centers is 13 cm. So, we have our first equation:
\(r_1 + r_2 = 13 \quad (Equation\ 1)\)
Using the Area Information
The area of a circle with radius \(r\) is given by the formula \(\pi r^2\). The sum of the areas of the two circles is given as \(89\pi\) square cm.
So, the sum of their areas can be written as:
\(\pi r_1^2 + \pi r_2^2 = 89\pi\)
We can divide both sides of the equation by \(\pi\) to simplify it:
\(r_1^2 + r_2^2 = 89 \quad (Equation\ 2)\)
Finding the Difference in Radii
We have two equations:
- \(r_1 + r_2 = 13\)
- \(r_1^2 + r_2^2 = 89\)
We want to find the difference in their radii, which is \(|r_1 - r_2|\).
We can use the algebraic identity \((a+b)^2 = a^2 + b^2 + 2ab\) and \((a-b)^2 = a^2 + b^2 - 2ab\).
Let's square the first equation \((r_1 + r_2 = 13)\):
\((r_1 + r_2)^2 = 13^2\)
\(r_1^2 + r_2^2 + 2r_1 r_2 = 169\)
Now, substitute the value of \(r_1^2 + r_2^2\) from Equation 2 into this equation:
\(89 + 2r_1 r_2 = 169\)
Solve for \(2r_1 r_2\):
\(2r_1 r_2 = 169 - 89\)
\(2r_1 r_2 = 80\)
Now we can use the identity for \((r_1 - r_2)^2\):
\((r_1 - r_2)^2 = r_1^2 + r_2^2 - 2r_1 r_2\)
Substitute the values of \(r_1^2 + r_2^2\) (which is 89) and \(2r_1 r_2\) (which is 80) into this equation:
\((r_1 - r_2)^2 = 89 - 80\)
\((r_1 - r_2)^2 = 9\)
To find the difference in radii, take the square root of both sides:
\(|r_1 - r_2| = \sqrt{9}\)
\(|r_1 - r_2| = 3\)
The difference in their radii is 3 cm.
Summary of Calculations
We used the given information to form a system of equations and solved for the difference in radii using algebraic identities.
| Given Information | Mathematical Representation |
|---|---|
| Circles touch externally, distance between centers = 13 cm | \(r_1 + r_2 = 13\) |
| Sum of areas = \(89\pi\) sq cm | \(\pi r_1^2 + \pi r_2^2 = 89\pi \implies r_1^2 + r_2^2 = 89\) |
| Algebraic Identity 1 | \((r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2\) |
| Calculated from Identity 1 | \(13^2 = 89 + 2r_1 r_2 \implies 169 = 89 + 2r_1 r_2 \implies 2r_1 r_2 = 80\) |
| Algebraic Identity 2 (for difference) | \((r_1 - r_2)^2 = r_1^2 + r_2^2 - 2r_1 r_2\) |
| Calculated Difference Squared | \((r_1 - r_2)^2 = 89 - 80 = 9\) |
| Difference in Radii | \(|r_1 - r_2| = \sqrt{9} = 3\) cm |
The difference in the radii of the two externally touching circles is 3 cm.
Revision Table: Key Circle Concepts
| Concept | Formula / Property |
|---|---|
| Area of a Circle | \(A = \pi r^2\) |
| Circumference of a Circle | \(C = 2\pi r\) or \(C = \pi d\) |
| Circles Touching Externally | Distance between centers = Sum of radii (\(d = r_1 + r_2\)) |
| Circles Touching Internally | Distance between centers = Difference of radii (\(d = |r_1 - r_2|\)), provided one circle is inside the other. |
Additional Information: Solving for Individual Radii
While the problem only asked for the difference, we can also find the values of the individual radii using the equations \(r_1 + r_2 = 13\) and \(r_1 r_2 = 40\).
Consider \(r_1\) and \(r_2\) as the roots of a quadratic equation of the form \(x^2 - (r_1+r_2)x + r_1r_2 = 0\).
Substituting the values:
\(x^2 - 13x + 40 = 0\)
We can factor this quadratic equation:
\((x - 5)(x - 8) = 0\)
The possible values for \(x\) are 5 and 8. Therefore, the radii of the two circles are 5 cm and 8 cm.
Let's check:
- Sum of radii: \(5 + 8 = 13\) cm (Matches distance between centers)
- Sum of areas: \(\pi (5)^2 + \pi (8)^2 = 25\pi + 64\pi = 89\pi\) square cm (Matches given sum of areas)
- Difference in radii: \(|8 - 5| = 3\) cm (Matches our calculated difference)
This confirms our result for the difference in radii is correct.
Similar Questions
An equilateral triangle ABC and a scalene triangle DBC are inscribed in a circle on same side of the arc. what is ∠BDC equal to?
What is the length of QC ?
What is the relation between x and y ?
If y = 15, then what is ∠ACB equal to ?
If r is the radius of S and R is the radius of S2, then which one of the following is correct?
If m is the area of the circle S and n is the area of semi-circle S1, then which one of the following is correct?
What is the radius of the circle ?
What is the area of the minor segment ?
- What is the area of the major segment ?
What is the length of the chord a unit circle which subtends an angle 2θ at the centre, where θ < 45°?
Important Questions from Circles, Chords and Tangents
A chord 21 cm long is drawn in a circle of diameter 25 cm. The perpendicular distance of the chord from the centre is:
AB is a chord of a circle with centre O and P is any point on the circle. If ∠APB = 112°, then what is the measure of ∠OAB ?
In a circle with radius 5 cm, a chord is at a distance of 3 cm from the centre. The length of the chord is:
The distance between the centres of two circles is 24 cm. If the radius of the two circles are 4 cm and 8 cm, then what is the sum of the lengths (in cm) of the direct common tangent and the transverse common tangent?
An equilateral triangle ABC and a scalene triangle DBC are inscribed in a circle on same side of the arc. what is ∠BDC equal to?
