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.

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.