Given: A circle with center $O$ and radius $r = 5\text{ cm}$. The angle between the two tangents is $60^{\circ}$.

To Find: Construct the pair of tangents and provide the geometric justification for the construction.

Step 1: Conceptual Analysis
Let the circle have center $O$. Let the two tangents meet at point $P$. Let the points of contact be $A$ and $B$.
In the quadrilateral $OAPB$:
- $\angle OAP = 90^{\circ}$ (Radius is perpendicular to the tangent at the point of contact).
- $\angle OBP = 90^{\circ}$ (Radius is perpendicular to the tangent at the point of contact).
- $\angle APB = 60^{\circ}$ (Given).
- The sum of angles in a quadrilateral is $360^{\circ}$.
Therefore, $\angle AOB = 360^{\circ} - (90^{\circ} + 90^{\circ} + 60^{\circ}) = 360^{\circ} - 240^{\circ} = 120^{\circ}$.

Step 2: Construction Procedure
1. Draw a circle of radius $5\text{ cm}$ with center $O$.
2. Draw any radius $OA$.
3. Construct an angle of $120^{\circ}$ at the center $O$ such that $\angle AOB = 120^{\circ}$.
4. At point $A$, construct a perpendicular to $OA$.
5. At point $B$, construct a perpendicular to $OB$.
6. Let these two perpendiculars intersect at point $P$. $PA$ and $PB$ are the required tangents.

Step 3: Justification
In quadrilateral $OAPB$:
- $\angle OAP = 90^{\circ}$ [By construction, as the tangent is perpendicular to the radius].
- $\angle OBP = 90^{\circ}$ [By construction, as the tangent is perpendicular to the radius].
- $\angle AOB = 120^{\circ}$ [By construction].
- The sum of interior angles of a quadrilateral is $360^{\circ}$.
- $\angle APB = 360^{\circ} - (\angle OAP + \angle OBP + \angle AOB)$
- $\angle APB = 360^{\circ} - (90^{\circ} + 90^{\circ} + 120^{\circ})$
- $\angle APB = 360^{\circ} - 300^{\circ} = 60^{\circ}$.
Thus, the tangents are inclined to each other at $60^{\circ}$.

Final Answer: The construction is justified as the angle between the tangents is calculated to be $60^{\circ}$ based on the properties of the quadrilateral $OAPB$ formed by the radii and the tangents.

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 $4.47\text{ cm}$.

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.

Given: A right-angled triangle $ABC$ where $AB = 6$ cm, $BC = 8$ cm, and $\angle B = 90^{\circ}$. $BD$ is the altitude from $B$ to the hypotenuse $AC$. A circle is drawn passing through points $B$, $C$, and $D$.

To Find: Construct the tangents from point $A$ to the circle passing through $B$, $C$, and $D$, and provide the justification.

Step 1: Construction of Triangle ABC

1. Draw a line segment $BC = 8$ cm.

2. At point $B$, construct an angle of $90^{\circ}$ using a compass and ruler.

3. Cut an arc of $6$ cm on the perpendicular line from $B$ to mark point $A$.

4. Join $AC$. This forms the right-angled triangle $ABC$.

Step 2: Construction of the Circle

1. Draw a perpendicular from $B$ to $AC$ to locate point $D$. [Since $BD \perp AC$, $\angle BDC = 90^{\circ}$].

2. Since $\angle BDC = 90^{\circ}$, the segment $BC$ subtends a right angle at $D$. Therefore, $BC$ is the diameter of the circle passing through $B, D,$ and $C$.

3. Find the midpoint $O$ of $BC$.

4. With $O$ as the center and $OB$ as the radius, draw a circle. This circle passes through $B, D,$ and $C$.

Step 3: Construction of Tangents from A

1. Join $AO$.

2. Find the midpoint $M$ of $AO$ by drawing the perpendicular bisector of $AO$.

3. With $M$ as the center and $MA$ as the radius, draw a circle. This circle intersects the circle with center $O$ at two points. One point is $B$, and let the other point be $E$.

4. Join $AE$. $AB$ and $AE$ are the required tangents.

Justification:

1. $AB$ is already a tangent to the circle because $\angle ABO = 90^{\circ}$ (given in the triangle construction). Since $OB$ is the radius, a line perpendicular to the radius at its endpoint on the circle is a tangent.

2. For the second tangent $AE$: In the circle with center $M$, $\angle AEO = 90^{\circ}$ [Angle in a semicircle].

3. Since $OE$ is a radius of the circle with center $O$, and $AE \perp OE$ at point $E$ on the circle, $AE$ must be a tangent to the circle.

Final Answer: The tangents from point $A$ to the circle are the line segments $AB$ and $AE$, where $B$ is the vertex of the right angle and $E$ is the point of intersection of the auxiliary circle with the circle passing through $B, C, D$.

Given: A circle with center $O$ and radius $r = 3\text{ cm}$. A diameter is drawn and extended on both sides. Two points $P$ and $Q$ are marked on this extended diameter such that $OP = 7\text{ cm}$ and $OQ = 7\text{ cm}$.

To Find: Construct tangents from points $P$ and $Q$ to the circle and provide the geometric justification for the construction.

Steps of Construction:

Step 1: Draw a circle with center $O$ and radius $3\text{ cm}$.

Step 2: Draw a line passing through $O$ and mark points $P$ and $Q$ on this line such that $OP = 7\text{ cm}$ and $OQ = 7\text{ cm}$.

Step 3: Find the midpoint of $OP$. Let this be $M_1$. With $M_1$ as center and $M_1P$ as radius, draw a circle. This circle intersects the original circle at points $A$ and $B$.

Step 4: Join $PA$ and $PB$. These are the required tangents from $P$.

Step 5: Find the midpoint of $OQ$. Let this be $M_2$. With $M_2$ as center and $M_2Q$ as radius, draw a circle. This circle intersects the original circle at points $C$ and $D$.

Step 6: Join $QC$ and $QD$. These are the required tangents from $Q$.

Justification:

To justify the construction, consider the tangent $PA$. Join $OA$.

In $\triangle OAP$, the angle $\angle OAP$ is an angle in a semicircle (since the circle with diameter $OP$ passes through $A$).

[By Thales' Theorem/Angle in a semicircle property: An angle inscribed in a semicircle is a right angle.]

Therefore, $\angle OAP = 90^\circ$.

Since $OA$ is the radius of the original circle and $PA$ is a line segment perpendicular to the radius at its point of contact $A$, $PA$ must be a tangent to the circle.

[Theorem: A line drawn perpendicular to the radius at the point of contact is a tangent to the circle.]

The same logic applies to $PB$, $QC$, and $QD$.

Calculation of Tangent Length:

In right-angled triangle $\triangle OAP$:

$OP^2 = OA^2 + AP^2$ [Using Pythagoras Theorem]

$7^2 = 3^2 + AP^2$

$49 = 9 + AP^2$

$AP^2 = 40$

$AP = \sqrt{40} = 2\sqrt{10} \approx 6.32\text{ cm}$.

Since $OQ = OP = 7\text{ cm}$ and the radius $OA = OC = 3\text{ cm}$, by symmetry, the lengths of all tangents $PA, PB, QC, QD$ are equal to $\sqrt{40}\text{ cm}$.

Final Answer: The tangents $PA, PB, QC,$ and $QD$ have been constructed, each with a length of $\sqrt{40} \text{ cm} \approx 6.32 \text{ cm}$.

Given: A circle drawn using a bangle (the center $O$ of which is unknown) and a point $P$ located outside the circle.

To Find: Construct a pair of tangents from point $P$ to the circle and provide the geometric justification for the construction.

Visual Representation:

Step 1: Finding the Center of the Circle

Since the circle is drawn with a bangle, the center $O$ is unknown. To find it:

1. Draw two non-parallel chords $AB$ and $CD$ in the circle.

2. Construct the perpendicular bisectors of $AB$ and $CD$.

3. The point where these two perpendicular bisectors intersect is the center $O$ of the circle. [Justification: The perpendicular bisector of any chord passes through the center of the circle.]

Step 2: Construction of Tangents

1. Join the point $P$ to the center $O$ with a line segment $OP$.

2. Construct the perpendicular bisector of $OP$. Let the midpoint of $OP$ be $M$.

3. With $M$ as the center and $MO$ (or $MP$) as the radius, draw a circle. This circle will intersect the original circle at two points, say $Q$ and $R$.

4. Join $PQ$ and $PR$. These are the required tangents.

Step 3: Justification

To justify that $PQ$ and $PR$ are tangents, we must prove that $\angle OQP = 90^\circ$ and $\angle ORP = 90^\circ$.

1. Join $OQ$.

2. In the circle with diameter $OP$, $\angle OQP$ is an angle in a semicircle. [Theorem: An angle inscribed in a semicircle is a right angle.]

3. Therefore, $\angle OQP = 90^\circ$.

4. Since $OQ$ is a radius of the original circle, $PQ$ must be a tangent to the circle at $Q$. [Theorem: A line perpendicular to the radius at its point of contact is a tangent to the circle.]

5. By the same logic, joining $OR$ proves $\angle ORP = 90^\circ$, confirming $PR$ is also a tangent.

Final Answer: The construction is completed by locating the center $O$, bisecting $OP$, and drawing an auxiliary circle to find the points of tangency $Q$ and $R$. The lines $PQ$ and $PR$ are the required tangents.

Given: A line segment $AB$ of length $8\text{ cm}$. Two circles are drawn: one with center $A$ and radius $r_1 = 4\text{ cm}$, and another with center $B$ and radius $r_2 = 3\text{ cm}$.

To Find/Construct: Construct tangents to the circle centered at $A$ from point $B$, and tangents to the circle centered at $B$ from point $A$. Provide the geometric justification for the construction.

Step 1: Construction Procedure

1. Draw a line segment $AB = 8\text{ cm}$.

2. Draw a circle with center $A$ and radius $4\text{ cm}$.

3. Draw a circle with center $B$ and radius $3\text{ cm}$.

4. To construct tangents from $B$ to circle $A$: Find the midpoint $M$ of $AB$ by drawing the perpendicular bisector of $AB$.

5. With $M$ as the center and $MA$ as the radius, draw a circle. This circle intersects circle $A$ at points $P$ and $Q$.

6. Join $BP$ and $BQ$. These are the required tangents from $B$ to circle $A$.

7. To construct tangents from $A$ to circle $B$: Using the same midpoint $M$, draw a circle with radius $MA$. This circle intersects circle $B$ at points $R$ and $S$.

8. Join $AR$ and $AS$. These are the required tangents from $A$ to circle $B$.

Step 2: Justification

To justify the construction, consider the tangents from $B$ to circle $A$ (points $P$ and $Q$):

Join $AP$. In $\triangle APB$, $\angle APB$ is an angle in a semicircle. [By Thales' Theorem, the angle subtended by a diameter at the circumference is $90^\circ$].

Since $M$ is the midpoint of $AB$, $MA = MB = MP$ (radii of the circle with center $M$). Thus, $\angle APB = 90^\circ$.

Since $AP$ is a radius of circle $A$ and $\angle APB = 90^\circ$, it follows that $BP$ must be a tangent to circle $A$ at point $P$. [A line perpendicular to the radius at the point of contact is a tangent].

The same logic applies to $BQ$, $AR$, and $AS$.

Step 3: Verification of Tangent Lengths

In $\triangle APB$, by the Pythagorean theorem: $BP^2 = AB^2 - AP^2$.

$BP^2 = 8^2 - 4^2 = 64 - 16 = 48$.

$BP = \sqrt{48} = 4\sqrt{3} \approx 6.93\text{ cm}$.

In $\triangle ARB$, by the Pythagorean theorem: $AR^2 = AB^2 - BR^2$.

$AR^2 = 8^2 - 3^2 = 64 - 9 = 55$.

$AR = \sqrt{55} \approx 7.42\text{ cm}$.

Final Answer: The tangents have been constructed such that the angle between the radius and the tangent is $90^\circ$, confirming the geometric validity of the construction. The lengths of the tangents from $B$ to circle $A$ are $4\sqrt{3}\text{ cm}$ and from $A$ to circle $B$ are $\sqrt{55}\text{ cm}$.