In the given figure, PT is a tangent to the circle with centre O and radius r. If \(\angle\) POT = 45\(^{\circ}\), then the length of OP is :
Show Hint
An isosceles right-angled triangle (a \( 45^{\circ}-45^{\circ}-90^{\circ} \) triangle) always has its sides in the ratio \( 1 : 1 : \sqrt{2} \).
Since \( \angle OTP = 90^{\circ} \) and \( \angle POT = 45^{\circ} \), the remaining angle \( \angle OPT \) is also \( 45^{\circ} \).
Therefore, the two perpendicular sides are equal: \( OT = PT = r \).
The hypotenuse \( OP \) is always \( \sqrt{2} \) times the length of the equal sides: \( OP = r\sqrt{2} \).
This geometric rule lets you find the answer instantly without any trigonometry.
Step 1: Understanding the Question:
The topic of this question is Geometry (Properties of Tangents to Circles) and right-angled triangle trigonometry.
We are given a circle with center \( O \) and radius \( r \).
A tangent \( PT \) is drawn to the circle from an external point \( P \), meeting the circle at point \( T \).
We are given the angle \( \angle POT = 45^{\circ} \) and we need to determine the length of the segment \( OP \).
Step 2: Key Formula or Approach:
- The tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the radius \( OT \) is perpendicular to the tangent \( PT \), making \( \angle OTP = 90^{\circ} \).
- Consequently, \( \Delta OTP \) is a right-angled triangle with the right angle at vertex \( T \).
- We can use standard trigonometric ratios in this right-angled triangle to find the length of the hypotenuse \( OP \):
\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Step 3: Detailed Explanation:
1. In triangle \( \Delta OTP \), since the tangent \( PT \) is perpendicular to the radius \( OT \) at the point of contact \( T \), we have:
\[ \angle OTP = 90^{\circ} \]
This establishes that \( \Delta OTP \) is a right-angled triangle.
2. The side \( OT \) is the radius of the circle, so:
\[ OT = r \]
3. The angle \( \angle POT \) is given as \( 45^{\circ} \).
4. In the right-angled triangle \( \Delta OTP \), with respect to the angle \( \angle POT = 45^{\circ} \):
- The adjacent side is the radius \( OT \).
- The hypotenuse is the line segment \( OP \).
5. Write the cosine ratio for angle \( \angle POT \):
\[ \cos(\angle POT) = \frac{OT}{OP} \]
6. Substitute the known values of \( \angle POT = 45^{\circ} \) and \( OT = r \):
\[ \cos 45^{\circ} = \frac{r}{OP} \]
7. Recall the standard trigonometric value:
\[ \cos 45^{\circ} = \frac{1}{\sqrt{2}} \]
8. Substitute this value into the equation:
\[ \frac{1}{\sqrt{2}} = \frac{r}{OP} \]
9. Solve for the length of \( OP \) by cross-multiplication:
\[ OP = r\sqrt{2} \]
Step 4: Final Answer:
The length of OP is \(r\sqrt{2}\), which corresponds to option (A).
