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Q1:
Choose the correct option and give justification. From a point $Q$, the length of the tangent to a circle is $24$ cm and the distance of $Q$ from the centre is $25$ cm. The radius of the circle is
Choose the correct option and give justification. From a point $Q$, the length of the tangent to a circle is $24$ cm and the distance of $Q$ from the centre is $25$ cm. The radius of the circle is
Solution :
Given:
A circle with center $O$. A point $Q$ lies outside the circle such that the distance from the center $O$ to point $Q$ is $OQ = 25$ cm. A tangent is drawn from $Q$ to the circle, touching the circle at point $P$, such that the length of the tangent $QP = 24$ cm.
To Find:
The radius of the circle, denoted as $OP$.
Step 1: Applying the Tangent-Radius Theorem
According to the theorem: "The tangent at any point of a circle is perpendicular to the radius through the point of contact."
Therefore, $OP \perp QP$. This implies that $\angle OPQ = 90^\circ$.
Step 2: Identifying the Triangle
Since $\angle OPQ = 90^\circ$, the triangle $\triangle OPQ$ is a right-angled triangle, where $OQ$ is the hypotenuse, and $OP$ and $QP$ are the legs of the triangle.
Step 3: Applying the Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. [Pythagorean Theorem: $a^2 + b^2 = c^2$]
Substituting the known values into the equation:
$OP^2 + QP^2 = OQ^2$
$OP^2 + (24)^2 = (25)^2$
Step 4: Algebraic Calculation
Calculate the squares of the given lengths:
$OP^2 + 576 = 625$
Isolate $OP^2$ by subtracting 576 from both sides:
$OP^2 = 625 - 576$
$OP^2 = 49$
Take the square root of both sides:
$OP = \sqrt{49}$
$OP = 7$ cm
Conclusion:
The radius of the circle $OP$ is $7$ cm.
Final Answer: 7 cm
More Questions from Class 10 Mathematics Circles EXERCISE 10.2
- Q10: Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
- Q11: Prove that the parallelogram circumscribing a circle is a rhombus.
- Q12: A triangle $ABC$ is drawn to circumscribe a circle of radius $4$ cm such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $8$ cm and $6$ cm respectively (see Fig. 10.14). Find the sides $AB$ and $AC$.
- Q13: Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
- Q2: Choose the correct option and give justification. In Fig. 10.11, if $TP$ and $TQ$ are the two tangents to a circle with centre $O$ so that $\angle POQ = 110^{\circ}$, then $\angle PTQ$ is equal to
- Q3: Choose the correct option and give justification. If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at angle of $80^{\circ}$, then $\angle POA$ is equal to
- Q4: Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
- Q5: Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
- Q6: The length of a tangent from a point $A$ at distance $5$ cm from the centre of the circle is $4$ cm. Find the radius of the circle.
- Q7: Two concentric circles are of radii $5$ cm and $3$ cm. Find the length of the chord of the larger circle which touches the smaller circle.
- Q8: A quadrilateral $ABCD$ is drawn to circumscribe a circle (see Fig. 10.12). Prove that $AB + CD = AD + BC$.
- Q9: In Fig. 10.13, $XY$ and $X'Y'$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $X'Y'$ at $B$. Prove that $\angle AOB = 90^{\circ}$.
CBSE Solutions for Class 10 Mathematics Circles
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Circles
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20 yrs of
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Teaching is my Passion. I have taught Mathematics, Physics and Chemistry to several students during 20+ years of my teaching. Many of my students who initially were scarred of Mathematics, have developed interest on the subject and opted MPC group in their Intermediate and are now earning Dollars as Software Engineers in USA, CANADA, UK and Australia.
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I have a total of 10 years of teaching experience. I am Teaching Online for Overseas and Local Students. My Students are from INDIA, UK, USA, INDONESIA & THAILAND. My passion is Teaching. I am Focusing on CBSE, IGCSE, IB, ICSE,&NIOS,.I have conducted National level conference and workshop, Hands-on-training, Seminar, etc Teaching with Human touch produces a great result. My students of class 10th had proven with good results.
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