The length of a tangent drawn from a point 10 cm away from the centre of the circle of radius 5 cm is

Options

A. 5 cm
B. `5 sqrt(3)` cm
C. `2 sqrt(3)` cm
D. `sqrt(15)` cm

Answer

To find the length of the tangent drawn from a point 10 cm away from the center of a circle with a radius of 5 cm, we can use the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Identify the elements of the problem
- Let O be the center of the circle.
- Let A be the point from which the tangent is drawn, which is 10 cm away from O.
- Let T be the point where the tangent touches the circle.
- The radius OT of the circle is 5 cm.

### Step 2: Draw the right triangle
- The triangle OAT is a right triangle where:
- OA is the hypotenuse (10 cm).
- OT is one leg (the radius, which is 5 cm).
- AT is the other leg (the length of the tangent we need to find).

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ OA^2 = OT^2 + AT^2 \]
Substituting the known values:
\[ 10^2 = 5^2 + AT^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 100 = 25 + AT^2 \]

### Step 5: Rearrange the equation
Now, rearranging the equation to solve for AT²:
\[ AT^2 = 100 - 25 \]
\[ AT^2 = 75 \]

### Step 6: Solve for AT
Taking the square root of both sides:
\[ AT = \sqrt{75} \]
This can be simplified:
\[ AT = \sqrt{25 \times 3} = 5\sqrt{3} \text{ cm} \]

### Final Answer
The length of the tangent AT is \(5\sqrt{3}\) cm.

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