Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Question
the following diagram shows the chord [AB] in a circle of radius 8cm, where AB =12cm. find the shaded segment.
**Understanding Semi-Circles and Angle Bisectors** The diagram illustrates a semi-circle with points A and B on the circumference, acting as endpoints of the diameter. ### Key Elements of the Diagram: 1. **Semi-Circle:** The shape is half of a full circle. 2. **Diameter:** Line AB represents the diameter of the semi-circle, stretching from point A on the left to point B on the right. 3. **Angle Bisector:** A line is drawn from the midpoint of the diameter passing through the center of the semi-circle, dividing it into two equal parts. This creates two right-angled triangles within the semi-circle. 4. **Shaded Region:** The lower part of the semi-circle is shaded in purple, denoting specific interest, possibly for calculations or area considerations. ### Points of Interest: - **A and B:** These points are the endpoints of the diameter. - **Midpoint of Diameter:** This point is equidistant from both A and B and is crucial in understanding properties like symmetry in the semi-circle. ### Mathematical Implications: - **Radius:** The distance from the midpoint of the diameter to points A or B represents the radius of the semi-circle. - **Right-Angled Triangles:** The semi-circle forms two right-angled triangles when bisected through the center. The diagram allows us to explore various geometric principles such as the properties of angles in a semi-circle, the relationship between the radius and the diameter, and area calculations of specific regions within a geometric shape.
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Transcribed Image Text:**Understanding Semi-Circles and Angle Bisectors** The diagram illustrates a semi-circle with points A and B on the circumference, acting as endpoints of the diameter. ### Key Elements of the Diagram: 1. **Semi-Circle:** The shape is half of a full circle. 2. **Diameter:** Line AB represents the diameter of the semi-circle, stretching from point A on the left to point B on the right. 3. **Angle Bisector:** A line is drawn from the midpoint of the diameter passing through the center of the semi-circle, dividing it into two equal parts. This creates two right-angled triangles within the semi-circle. 4. **Shaded Region:** The lower part of the semi-circle is shaded in purple, denoting specific interest, possibly for calculations or area considerations. ### Points of Interest: - **A and B:** These points are the endpoints of the diameter. - **Midpoint of Diameter:** This point is equidistant from both A and B and is crucial in understanding properties like symmetry in the semi-circle. ### Mathematical Implications: - **Radius:** The distance from the midpoint of the diameter to points A or B represents the radius of the semi-circle. - **Right-Angled Triangles:** The semi-circle forms two right-angled triangles when bisected through the center. The diagram allows us to explore various geometric principles such as the properties of angles in a semi-circle, the relationship between the radius and the diameter, and area calculations of specific regions within a geometric shape.
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