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Q4:
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Solution :
Given: A circle with center $O$. Let $AB$ be a diameter of the circle. Let $PQ$ be a tangent at point $A$ and $RS$ be a tangent at point $B$.
To Prove: $PQ \parallel RS$.
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."
Since $PQ$ is a tangent at $A$ and $OA$ is the radius, we have:
$OA \perp PQ \implies \angle OAP = 90^\circ$ and $\angle OAQ = 90^\circ$.
Similarly, since $RS$ is a tangent at $B$ and $OB$ is the radius, we have:
$OB \perp RS \implies \angle OBR = 90^\circ$ and $\angle OBS = 90^\circ$.
Step 2: Analyzing the Angles
Consider the lines $PQ$ and $RS$ intersected by the transversal $AB$.
From Step 1, we have:
$\angle OAP = 90^\circ$
$\angle OBS = 90^\circ$
Therefore, $\angle OAP = \angle OBS = 90^\circ$.
Step 3: Establishing Parallelism
Observe that $\angle OAP$ and $\angle OBS$ are alternate interior angles with respect to lines $PQ$ and $RS$ and transversal $AB$.
[Theorem: If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel.]
Since $\angle OAP = \angle OBS = 90^\circ$, the alternate interior angles are equal.
Thus, $PQ \parallel RS$.
Final Answer: Since the alternate interior angles formed by the transversal $AB$ with lines $PQ$ and $RS$ are equal ($90^\circ$), the tangents $PQ$ and $RS$ are parallel to each other.
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
- 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
- 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 $\angle AOB = 90^{\circ}$.
- 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.
CBSE Solutions for Class 10 Mathematics Circles
Chapters in CBSE - Class 10 Mathematics
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