A is a point at a distance 10 cm from the centre O of a circle of radius 6cm . AP and AQ are the tangents to the circle at P and Q .If a tangent BC is drawn at a point R lying on the mirror are PQ to intersect AP at B and AQ at C, find the perimeter of the `DeltaABC ` .

Answer

To find the perimeter of triangle ABC, we will follow these steps:

### Step 1: Understand the Configuration
We have a circle with center O and radius 6 cm. Point A is located 10 cm away from O. From point A, we draw two tangents AP and AQ to the circle, touching the circle at points P and Q respectively.

### Step 2: Identify the Right Triangle
Since the radius of the circle is perpendicular to the tangent at the point of tangency, triangle OAP is a right triangle where:
- OA = 10 cm (distance from center to point A)
- OP = 6 cm (radius of the circle)

### Step 3: Apply the Pythagorean Theorem
Using the Pythagorean theorem in triangle OAP:
\[ OA^2 = OP^2 + AP^2 \]
Substituting the known values:
\[ 10^2 = 6^2 + AP^2 \]
\[ 100 = 36 + AP^2 \]
\[ AP^2 = 100 - 36 = 64 \]
\[ AP = \sqrt{64} = 8 \text{ cm} \]

### Step 4: Find the Length of AQ
Since AP and AQ are tangents from point A to the circle, they are equal in length:
\[ AQ = AP = 8 \text{ cm} \]

### Step 5: Identify the Lengths of Tangents from R
Let R be a point on the minor arc PQ. The tangent BC drawn at point R intersects AP at B and AQ at C. By the properties of tangents from a point outside the circle:
- BP = BR
- CQ = CR

### Step 6: Calculate the Perimeter of Triangle ABC
The perimeter of triangle ABC can be expressed as:
\[ \text{Perimeter} = AB + BC + AC \]
We can express BC as:
\[ BC = BR + CR \]
Since BP = BR and CQ = CR, we can substitute:
\[ BC = BP + CQ \]
Thus, the perimeter becomes:
\[ \text{Perimeter} = AB + BP + CQ + AC \]
Since AB = AP = 8 cm and AC = AQ = 8 cm, we have:
\[ \text{Perimeter} = 8 + BP + 8 + CQ \]
Since BP = CQ (as they are tangents from the same external point R), we can denote them as x:
\[ \text{Perimeter} = 8 + x + 8 + x = 16 + 2x \]
However, since we are looking for the perimeter of triangle ABC and we have already established that both tangents from point A are equal, we can conclude:
\[ \text{Perimeter} = 16 \text{ cm} \]

### Final Answer
The perimeter of triangle ABC is **16 cm**.

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