The Length of a Tangent from a Point A at distance 5cm from the Centre of the Circle is 4cm. Find the Radius of the Circle

The Length of a Tangent from a Point A at distance 5cm from the Centre of the Circle is 4cm. Find the Radius of the Circle

To find the radius of a circle when given the length of a tangent from a point, use the relationship between tangents and radii in a circle.

The length of the tangent from point A to the circle's center is 4 cm (AT = 4 cm).

The distance between point A and the circle's center is 5 cm (OA = 5 cm).

A tangent drawn from an external point to a circle is perpendicular to the radius at the point of contact (T).

This means that OT ⊥ AT, forming a right triangle OTA.

Using Pythagoras' Theorem:

In a right triangle, the square of the hypotenuse (OA) is equal to the sum of the squares of the other two sides (OT and AT). This is represented by the Pythagorean equation:

OA² = OT² + AT²

Substitute the given values into the Pythagorean theorem equation

5² = OT² + 4²

25 − 16 = OT²

OT = 3 cm

Since OT represents the radius of the circle, the radius is also 3 cm.

Therefore, the radius of the circle is 3 cm, as determined by the length of the tangent from point A.

Tangents and Circles

A tangent is a line that touches a circle at exactly one point, known as the point of tangency. When a tangent is drawn from an external point to a circle, it forms a right angle with the radius at the point of tangency. The radius of a circle is the distance from the center of the circle to any point on the circle's circumference.

Using Pythagoras' Theorem:

• In a right triangle, Pythagoras' theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
• For a circle, if we know the length of the tangent from an external point to the point of tangency and the distance from the external point to the circle's center, we can use Pythagoras' theorem to find the radius.

Applying the Theorem:

Let's say the length of the tangent from the external point to the point of tangency is 't' units, and the distance from the external point to the circle's center is 'd' units.

Using Pythagoras' theorem, we set up the equation d² = t² + r², where 'r' is the radius we want to find.

Solving for 'r', we get r = √d² - t² which gives us the radius of the circle.

By substituting the known values of 't' and 'd' into the equation, we can calculate the radius of the circle.

This method is useful in geometry and trigonometry to determine circle properties and relationships between tangents, radii, and external points.