The length of the tangent drawn to a circle of radius 4 cm from a point 5 cm away from the centre of the circle is

Options

A. 3cm
B. `4 sqrt(2)` cm
C. `5 sqrt(2) cm`
D. `3 sqrt(2)` cm

Answer

To find the length of the tangent drawn to a circle from a point outside the circle, we can use the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Identify the given values
- The radius of the circle (r) = 4 cm
- The distance from the external point to the center of the circle (d) = 5 cm

### Step 2: Understand the relationship in the triangle
When a tangent is drawn from an external point to a circle, it forms a right triangle with:
- One leg as the radius (4 cm)
- The other leg as the tangent length (which we need to find)
- The hypotenuse as the distance from the external point to the center of the circle (5 cm)

### Step 3: Set up the Pythagorean theorem
According to the Pythagorean theorem:
\[ d^2 = r^2 + t^2 \]
Where:
- \(d\) is the distance from the external point to the center of the circle (5 cm)
- \(r\) is the radius of the circle (4 cm)
- \(t\) is the length of the tangent we want to find

### Step 4: Substitute the known values into the equation
\[ 5^2 = 4^2 + t^2 \]
This simplifies to:
\[ 25 = 16 + t^2 \]

### Step 5: Solve for \(t^2\)
Subtract 16 from both sides:
\[ 25 - 16 = t^2 \]
\[ 9 = t^2 \]

### Step 6: Take the square root to find \(t\)
\[ t = \sqrt{9} = 3 \text{ cm} \]

### Conclusion
The length of the tangent drawn to the circle from the point 5 cm away from the center is **3 cm**.

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