Question

If radii of two concentric circles are \[4\text{ }cm\] and \[\text{5 }cm\] then the length of each chord of one circle which is tangent to the other circle is:
A). \[\text{3 }cm\]
B). \[\text{6 }cm\]
C). \[\text{9 }cm\]
D). \[\text{1 }cm\]

Answer 1

Hint: In this question firstly describe the radius of a small circle and the radius of a large circle and then with the help of a given statement make the figure. After that, apply the Pythagoras theorem to find the base and then double the base to find the value of the chord and check which option is correct in the given options.

Complete step-by-step solution:
A circle is a flat figure with a curved surface. Every point on the circle is equidistant from a single point known as the circle's centre. A circle is a two-dimensional form with a radius measurement.
If two or more things share a similar centre, they are said to be concentric in geometry. Concentric circles, spheres, regular polyhedra, and regular polygons all have the same centre point. Two concentric circles in Euclidean Geometry have the same centre but always have distinct radii. Concentric circles have the same or a similar centre. Concentric circles, in other words, are defined as two or more circles that share the same centre point.
A circle is a two-dimensional shape created by a set of points in the plane that are separated by a constant or fixed distance. The origin or centre of the circle is the fixed point, and the radius is the fixed distance between the points from the origin.
A circle's chord is any line segment that connects two locations on the circle's circumference. The circle is divided into two pieces by the chord.
In the question we have given two concentric circles of radius \[4\text{ }cm\] and \[\text{5 }cm\]
Where radius of small circle is \[4\text{ }cm\]
Radius of large circle is \[\text{5 }cm\]

Therefore we will find out the length of chord of one circle that is tangent to the other circle
As shown in figure \[(1)\] it forms a right angle triangle therefore apply Pythagoras theorem:
\[\Rightarrow Hypotensus{{e}^{2}}=bas{{e}^{2}}+perpendicula{{r}^{2}}\]
\[\Rightarrow O{{T}^{2}}=P{{T}^{2}}+O{{P}^{2}}\]
\[\Rightarrow {{5}^{2}}=P{{T}^{2}}+{{4}^{2}}\]
\[\Rightarrow 25=P{{T}^{2}}+16\]
\[\Rightarrow P{{T}^{2}}=25-16\]
\[\Rightarrow P{{T}^{2}}=9\]
\[\Rightarrow PT=\sqrt{9}\]
\[\Rightarrow PT=3\text{ }cm\]
Therefore the length of the chord \[QT\] is given by:
\[\Rightarrow QT=PT\times 2\]
\[\Rightarrow QT=3\times 2\]
\[\Rightarrow QT=6\text{ }cm\]
Hence we can say that option \[(B)\] is correct.

Note: Students must remember to include units at the conclusion. To proceed with the solution, you must first understand the circle theorems, you must make diagrams using given data so that you can solve the question easily. Only when one of the triangle's angles is \[90\] degrees then Pythagoras theorem can be applied.