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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.
Solution :
Given: Two concentric circles with center $O$. The radius of the inner circle is $r_1 = 4\text{ cm}$ and the radius of the outer circle is $r_2 = 6\text{ cm}$.
To Find: Construct a tangent from a point $P$ on the outer circle to the inner circle, measure its length, and verify the result using the Pythagorean theorem.
Step 1: Construction Procedure
1. Draw a circle with center $O$ and radius $4\text{ cm}$.
2. Draw a concentric circle with center $O$ and radius $6\text{ cm}$.
3. Take a point $P$ on the outer circle. Join $OP$.
4. Find the midpoint $M$ of $OP$ by drawing the perpendicular bisector of $OP$.
5. With $M$ as the center and $MO$ as the radius, draw a circle. Let this circle intersect the inner circle at points $T$ and $T'$.
6. Join $PT$ and $PT'$. These are the required tangents.
Step 2: Justification
Join $OT$. Since $OT$ is the radius of the inner circle and $PT$ is the tangent, $\angle OTP = 90^\circ$ [Since the tangent at any point of a circle is perpendicular to the radius through the point of contact].
In $\triangle OTP$, by the Pythagorean theorem:
$OP^2 = OT^2 + PT^2$
Step 3: Calculation and Verification
Given $OP = 6\text{ cm}$ (radius of the outer circle) and $OT = 4\text{ cm}$ (radius of the inner circle).
Substituting the values into the Pythagorean equation:
$6^2 = 4^2 + PT^2$
$36 = 16 + PT^2$
$PT^2 = 36 - 16$
$PT^2 = 20$
$PT = \sqrt{20} = 2\sqrt{5}\text{ cm}$
Using $\sqrt{5} \approx 2.236$:
$PT \approx 2 \times 2.236 = 4.47\text{ cm}$
Final Answer: The length of the tangent is $2\sqrt{5}\text{ cm}$ or approximately $4.47\text{ cm}$.
More Questions from Class 10 Mathematics Constructions EXERCISE 11.2
- 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.
- 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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