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Q1:
In each of the following, give also the justification of the construction:
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.
Solution :
Given: A circle with center $O$ and radius $r = 6\text{ cm}$. A point $P$ located at a distance $OP = 10\text{ cm}$ from the center $O$.
To Find: Construct a pair of tangents from point $P$ to the circle and measure their lengths.
Step 1: Construction Procedure
1. Draw a circle with center $O$ and radius $6\text{ cm}$.
2. Mark a point $P$ such that $OP = 10\text{ cm}$.
3. Draw the line segment $OP$. Find the midpoint $M$ of $OP$ by drawing the perpendicular bisector of $OP$.
4. With $M$ as the center and $MO$ as the radius, draw a circle. This circle will intersect the original circle at two points, $T_1$ and $T_2$.
5. Join $PT_1$ and $PT_2$. These are the required tangents.
Step 2: Justification
To justify that $PT_1$ and $PT_2$ are tangents, we join $OT_1$.
In $\triangle OT_1P$, $\angle OT_1P$ is an angle in a semicircle. [By Thales' Theorem, the angle subtended by a diameter at the circumference is $90^\circ$].
Therefore, $\angle OT_1P = 90^\circ$.
Since $OT_1$ is a radius of the circle, $PT_1$ must be a tangent to the circle at $T_1$. [Since a line perpendicular to the radius at the point of contact is a tangent].
Step 3: Calculation of Tangent Length
In the right-angled triangle $\triangle OT_1P$:
$OP^2 = OT_1^2 + PT_1^2$ [Using Pythagoras Theorem]
Given $OP = 10\text{ cm}$ and $OT_1 = 6\text{ cm}$ (radius).
$10^2 = 6^2 + PT_1^2$
$100 = 36 + PT_1^2$
$PT_1^2 = 100 - 36$
$PT_1^2 = 64$
$PT_1 = \sqrt{64} = 8\text{ cm}$.
Final Answer: The length of each tangent is 8 cm.
More Questions from Class 10 Mathematics Constructions EXERCISE 11.2
- Q2: In each of the following, give also the justification of the construction: Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.
- Q3: In each of the following, give also the justification of the construction: Draw a circle of radius 3 cm. Take two points $P$ and $Q$ on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points $P$ and $Q$.
- Q4: In each of the following, give also the justification of the construction: Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of $60^{\circ}$.
- Q5: In each of the following, give also the justification of the construction: Draw a line segment $AB$ of length 8 cm. Taking $A$ as centre, draw a circle of radius 4 cm and taking $B$ as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.
- Q6: In each of the following, give also the justification of the construction: Let ABC be a right triangle in which $AB = 6$ cm, $BC = 8$ cm and $\angle B = 90^{\circ}$. $BD$ is the perpendicular from $B$ on $AC$. The circle through $B$, $C$, $D$ is drawn. Construct the tangents from $A$ to this circle.
- Q7: In each of the following, give also the justification of the construction: Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circle.
CBSE Solutions for Class 10 Mathematics Constructions
Chapters in CBSE - Class 10 Mathematics
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