UPSKILL MATH PLUS

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Result $$1$$:
A tangent at any point on a circle and the radius through the point are perpendicular to each other.
Explanation:

The tangent at the point $$P$$ on a circle and the radius through the point $$P$$ are perpendicular.

That is, the radius $$OP$$ makes an angle $$90^{\circ}$$ with the tangent $$AB$$ at the point $$P$$.
Example:
In the above given figure if $$OP$$ $$=$$ $$3$$ $$cm$$  and $$PQ$$ $$=$$ $$4$$ $$cm$$, find the length of $$OQ$$.

Solution:

By the result, $$\angle OPQ$$ $$=$$ $$90^{\circ}$$.

So, $$OPQ$$ is a right-angled triangle.

By the Pythagoras theorem, we have:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
$$OQ^2$$ $$=$$ $$OP^2$$ $$+$$ $$PQ^2$$

$$OQ^2$$ $$=$$ $$3^2$$ $$+$$ $$4^2$$

$$OQ^2$$ $$=$$ $$9 + 16$$

$$OQ^2$$ $$=$$ $$25$$

$$\Rightarrow$$  $$OQ$$ $$=$$ $$\sqrt{25}$$

$$OQ$$ $$=$$ $$5$$

Therefore, the measure of $$OQ$$ $$=$$ $$5$$ $$cm$$
Result $$2$$:
• No tangent can be drawn from an interior point of the circle.
• Only one tangent can be drawn at any point on a circle.
• Two tangents can be drawn from any exterior point of a circle.