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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.
Solution :
Given:
1. A circle with center $O$.
2. A point $A$ outside the circle such that the distance from the center $O$ to point $A$ is $OA = 5$ cm.
3. A tangent $AB$ from point $A$ to the circle at point $B$, where the length of the tangent $AB = 4$ cm.
To find:
The radius of the circle, denoted as $OB$.
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, $OB \perp AB$. This implies that $\angle OBA = 90^\circ$.
Step 2: Identifying the Geometric Figure
Since $\angle OBA = 90^\circ$, the triangle $\triangle OBA$ is a right-angled triangle, where $OA$ is the hypotenuse, $OB$ is the radius (base/height), and $AB$ is the tangent.
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. [Using Pythagoras Theorem: $Hypotenuse^2 = Base^2 + Height^2$]
$OA^2 = OB^2 + AB^2$
Step 4: Substituting the Given Values
Given $OA = 5$ cm and $AB = 4$ cm.
$(5)^2 = OB^2 + (4)^2$
Step 5: Solving for the Radius ($OB$)
$25 = OB^2 + 16$
Subtract 16 from both sides:
$OB^2 = 25 - 16$
$OB^2 = 9$
Taking the square root on both sides:
$OB = \sqrt{9}$
$OB = 3$ cm
[Since length cannot be negative, we take the positive root.]
Final Answer: The radius of the circle is 3 cm.
More Questions from Class 10 Mathematics Circles EXERCISE 10.2
- 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
- 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.
- 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
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