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Q3:
A tangent $PQ$ at a point $P$ of a circle of radius $5$ cm meets a line through the centre $O$ at a point $Q$ so that $OQ = 12$ cm. Length $PQ$ is :
A tangent $PQ$ at a point $P$ of a circle of radius $5$ cm meets a line through the centre $O$ at a point $Q$ so that $OQ = 12$ cm. Length $PQ$ is :
Solution :
Given:
- A circle with center $O$ and radius $OP = 5$ cm.
- A tangent $PQ$ at point $P$ on the circle.
- A line segment $OQ$ passing through the center $O$ such that $OQ = 12$ cm.
To find:
The length of the tangent $PQ$.
Step 1: Identifying the geometric relationship
According to the Theorem 10.1 of circles: "The tangent at any point of a circle is perpendicular to the radius through the point of contact."
Therefore, $OP \perp PQ$. This implies that $\angle OPQ = 90^\circ$.
Step 2: Applying the Pythagorean Theorem
Since $\triangle OPQ$ is a right-angled triangle with the right angle at $P$, we can apply the Pythagorean Theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In $\triangle OPQ$:
$OQ^2 = OP^2 + PQ^2$
Step 3: Substituting the known values
Given $OP = 5$ cm and $OQ = 12$ cm, we substitute these into the equation:
$(12)^2 = (5)^2 + PQ^2$
$144 = 25 + PQ^2$
Step 4: Solving for $PQ$
Subtract $25$ from both sides of the equation:
$PQ^2 = 144 - 25$
$PQ^2 = 119$
Taking the square root of both sides:
$PQ = \sqrt{119}$ cm
Final Answer: The length of the tangent $PQ$ is $\sqrt{119}$ cm.
More Questions from Class 10 Mathematics Circles EXERCISE 10.1
- Q1: How many tangents can a circle have?
- Q2(i): Fill in the blanks : (i) A tangent to a circle intersects it in __________ point (s).
- Q2(ii): Fill in the blanks : (ii) A line intersecting a circle in two points is called a __________.
- Q2(iii): Fill in the blanks : (iii) A circle can have __________ parallel tangents at the most.
- Q2(iv): Fill in the blanks : (iv) The common point of a tangent to a circle and the circle is called __________.
- Q4: Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
CBSE Solutions for Class 10 Mathematics Circles
Chapters in CBSE - Class 10 Mathematics
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