AB is the diameter of a circle with centre O. P be a point on it. If ∠AOP = 95°, then ∠OBP __________.
47.5°
Let's break down this geometry problem involving a circle, its diameter, and angles.
We are given a circle with centre O, where AB is the diameter and P is a point on the circle. We know that the angle ∠AOP is 95°. Our goal is to find the measure of ∠OBP.
Here's how we can approach this problem:
- Identify the properties of the given elements:
- O is the centre of the circle.
- AB is the diameter passing through O.
- P is a point on the circumference.
- OA, OB, and OP are all radii of the same circle. Therefore, OA = OB = OP.
- Consider the angles on the straight line AB:
- Since AB is a diameter, A, O, and B are collinear.
- The angles ∠AOP and ∠BOP form a linear pair.
- The sum of angles in a linear pair is 180°. So, ∠AOP + ∠BOP = 180°.
- We are given ∠AOP = 95°.
- Therefore, ∠BOP = 180° - ∠AOP = 180° - 95° = 85°.
- Focus on triangle OBP:
- We know that OB and OP are radii, so OB = OP.
- This means that triangle OBP is an isosceles triangle.
- In an isosceles triangle, the angles opposite the equal sides are equal. The angles opposite sides OB and OP are ∠OPB and ∠OBP, respectively.
- Thus, ∠OBP = ∠OPB.
- Calculate the angles in triangle OBP:
- The sum of angles in any triangle is 180°.
- In triangle OBP, ∠BOP + ∠OBP + ∠OPB = 180°.
- We found ∠BOP = 85°, and we know ∠OBP = ∠OPB.
- Let ∠OBP = ∠OPB = x.
- So, 85° + x + x = 180°.
- 85° + 2x = 180°.
- 2x = 180° - 85°.
- 2x = 95°.
- x = 95° / 2.
- x = 47.5°.
Therefore, ∠OBP = 47.5°.
Here is a summary of the angle calculations:
| Angle | Calculation | Value |
|---|---|---|
| ∠AOP | Given | 95° |
| ∠BOP | 180° - ∠AOP | $180° - 95° = 85°$ |
| ∠OBP & ∠OPB | $(180° - ∠BOP) / 2$ (in isosceles ▵OBP) | $(180° - 85°) / 2 = 95° / 2 = 47.5°$ |
The value of ∠OBP is 47.5°.
Revision Table: Key Concepts in Circle Geometry
Understanding basic circle geometry properties is crucial for solving problems like this. Here's a quick revision:
- Radius: A line segment from the center to any point on the circle. All radii of the same circle are equal.
- Diameter: A line segment passing through the center with endpoints on the circle. It's the longest chord and is twice the radius.
- Angles on a Straight Line: Angles that form a straight line add up to 180° (linear pair).
- Isosceles Triangle: A triangle with two equal sides. The angles opposite the equal sides are also equal.
- Sum of Angles in a Triangle: The sum of the interior angles of any triangle is always 180°.
Additional Information: Properties of Triangles in a Circle
When points on a circle are connected to the center or other points on the circle, triangles are often formed. Recognizing the type of triangle and its properties is key:
- Triangle formed by two radii and a chord: This is always an isosceles triangle because the two sides that are radii are equal. For example, triangle OBP in this problem is formed by two radii (OB, OP) and a chord (BP).
- Triangle inscribed in a semicircle: A triangle with its diameter as one side and the third vertex on the circumference always has a right angle at the vertex on the circumference. For example, ▵APB would be a right-angled triangle at P if ∠APB was the angle we were looking for (which it isn't in this problem).
Applying these fundamental geometric principles allows us to calculate unknown angles within the circle setup.
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