If two circles of radii 18 cm and 8 cm touch externally, then the length of a direct common tangent is:
24 cm
Let's find the length of the direct common tangent between two circles that touch externally. We are given the radii of the two circles.
Understanding Direct Common Tangents
A direct common tangent is a line segment that is tangent to both circles and on the same side of the line joining the centers of the circles. When two circles touch externally, the distance between their centers is equal to the sum of their radii.
Given Information:
- Radius of the first circle, \(r_1 = 18\) cm
- Radius of the second circle, \(r_2 = 8\) cm
Since the circles touch externally, the distance between their centers (\(d\)) is:
\(d = r_1 + r_2\)
\(d = 18 \text{ cm} + 8 \text{ cm} = 26 \text{ cm}\)
Formula for Direct Common Tangent Length
The length of a direct common tangent (\(L\)) between two circles with radii \(r_1\) and \(r_2\) and distance between centers \(d\) is given by the formula:
\(L = \sqrt{d^2 - (r_1 - r_2)^2}\)
Calculating the Length
We can substitute the values we have into the formula. Since the circles touch externally, \(d = r_1 + r_2\).
So the formula becomes:
\(L = \sqrt{(r_1 + r_2)^2 - (r_1 - r_2)^2}\)
Using the algebraic identity \( (a+b)^2 - (a-b)^2 = 4ab \), this simplifies to:
\(L = \sqrt{4r_1 r_2}\)
\(L = 2\sqrt{r_1 r_2}\)
Now, substitute the given radii \(r_1 = 18\) cm and \(r_2 = 8\) cm:
\(L = 2\sqrt{18 \times 8}\)
\(L = 2\sqrt{144}\)
Since \(\sqrt{144} = 12\), we get:
\(L = 2 \times 12\)
\(L = 24\) cm
Alternatively, using the original formula with \(d = 26\), \(r_1 = 18\), \(r_2 = 8\):
\(r_1 - r_2 = 18 - 8 = 10\) cm
\(L = \sqrt{26^2 - 10^2}\)
\(L = \sqrt{676 - 100}\)
\(L = \sqrt{576}\)
To find \(\sqrt{576}\), we can note that \(20^2 = 400\) and \(30^2 = 900\). The number ends in 6, so the root must end in 4 or 6. Let's check \(24^2\): \(24 \times 24 = 576\).
\(L = 24\) cm
Both methods yield the same result. The length of the direct common tangent is 24 cm.
| Parameter | Value |
|---|---|
| Radius of first circle (\(r_1\)) | 18 cm |
| Radius of second circle (\(r_2\)) | 8 cm |
| Distance between centers (\(d\)) | \(r_1 + r_2 = 18 + 8 = 26\) cm (since they touch externally) |
| Length of direct common tangent (\(L\)) | \(\sqrt{d^2 - (r_1 - r_2)^2} = 24\) cm |
Revision Table: Circle Tangents
| Type of Tangent | Description | Formula for Length (\(L\)) | Condition on \(d\) |
|---|---|---|---|
| Direct Common Tangent | Tangent lies on the same side of the line joining centers. | \(L = \sqrt{d^2 - (r_1 - r_2)^2}\) | \(d > r_1 + r_2\) (for distinct circles) \(d = r_1 + r_2\) (for externally touching circles) |
| Transverse Common Tangent | Tangent lies on opposite sides of the line joining centers. | \(L = \sqrt{d^2 - (r_1 + r_2)^2}\) | \(d > r_1 + r_2\) |
Additional Information on Circles and Tangents
- When two circles touch internally, the distance between their centers is the difference of their radii, \(d = |r_1 - r_2|\). In this case, there is only one common tangent (a direct common tangent) at the point of contact.
- If one circle is completely inside another without touching, there are no common tangents.
- If two circles intersect at two distinct points, they have two direct common tangents but no transverse common tangents.
- If two circles are separate from each other, they have two direct common tangents and two transverse common tangents.
- A tangent to a circle is always perpendicular to the radius at the point of contact.
Similar Questions
A chord 21 cm long is drawn in a circle of diameter 25 cm. The perpendicular distance of the chord from the centre is:
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?
In a circle with centre O, an arc ABC subtends an angle of 132° at the center of the circle. Chord AB is produced to point P. Then ∠CBP is equal to∶
What can be the maximum number of common tangent which can be drawn to two non-intersecting circles?
There are two identical circles of radius 10 cm each. If the length of the direct common tangent is 26 cm, then what is the length (in cm) of the transverse common tangent?
AB is a chord in the minor segment of a circle with centre O. C is a point on the minor arc (between A and B). The tangents to the circle at A and B meet at a point P. If ∠ACB = 108°, then ∠APB is equal to:
AB is the chord of a circle such that AB = 10 cm. If the diameter of the circle is 20 cm, then the angle subtended by the chord at the centre is ________.
Select the INCORRECT statement with respect to the properties of a circle.
The radius of a circle is 5 cm. Calculate the length of a tangent drawn to this circle from a point at a distance of 10 cm from its centre.
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?
