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E D A P M 6 in. C B

E D A P M 6 in. C B

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 44E
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round answers to the nearest tenth when not a whole value answer: Find each length to the nearest tenth: MC, PM, and PD Then in the second figure Find the measure of each angle: Angle YXW, Angle YXZ, and Angle XYZ.
**Title: Understanding Pentagons** **Image Description and Analysis:** In the diagram provided, we have a regular pentagon labeled \(ABCDE\). At the center of the pentagon, point \(P\) is marked, which connects to all five vertices of the pentagon, dividing it into five isosceles triangles. **Key Labels and Dimensions:** - The vertices of the pentagon are \(A\), \(B\), \(C\), \(D\), and \(E\). - The line \(AB\) is indicated to have a length of 6 inches, implying that all sides of this regular pentagon are of equal length (6 inches each). - Point \(M\) is located at the midpoint of side \(DC\). - A dashed line connects \(P\) to \(M\) perpendicularly, forming a right angle (noted by a small red square). **Significant Geometric Observations:** 1. **Regular Pentagon Properties:** - It has five equal sides. - It has five equal interior angles. - The sum of the interior angles of a pentagon is \(540^\circ\). 2. **Isosceles Triangles Inside the Pentagon:** - Each of the five triangles, such as \( \triangle APB\), is an isosceles triangle with \( P \) being the apex and the base being one of the pentagon's sides. 3. **Perpendicular Bisectors:** - The line segment \( PM \) is perpendicular to the side \( DC \), indicating that \(M\) is not only the midpoint of \(DC\) but also on the axis of symmetry for the triangle \(PDC\). **Educational Insights:** - **Symmetry and Congruence:** The pentagon in this image is a great example to study symmetry and congruence, with point \(P\) acting as a central point from which equal paths radiate to all vertices. - **Central Angles:** Since \(P\) is the center, the angle subtended by any side of the pentagon at \(P\) will be equal, e.g., \(\angle APB\). - **Real-World Applications:** Understanding the properties of regular polygons can be helpful in various fields, including architecture, design, and even nature studies (e.g., pentagonal tiling and symmetry in flowers). By de
Transcribed Image Text:**Title: Understanding Pentagons** **Image Description and Analysis:** In the diagram provided, we have a regular pentagon labeled \(ABCDE\). At the center of the pentagon, point \(P\) is marked, which connects to all five vertices of the pentagon, dividing it into five isosceles triangles. **Key Labels and Dimensions:** - The vertices of the pentagon are \(A\), \(B\), \(C\), \(D\), and \(E\). - The line \(AB\) is indicated to have a length of 6 inches, implying that all sides of this regular pentagon are of equal length (6 inches each). - Point \(M\) is located at the midpoint of side \(DC\). - A dashed line connects \(P\) to \(M\) perpendicularly, forming a right angle (noted by a small red square). **Significant Geometric Observations:** 1. **Regular Pentagon Properties:** - It has five equal sides. - It has five equal interior angles. - The sum of the interior angles of a pentagon is \(540^\circ\). 2. **Isosceles Triangles Inside the Pentagon:** - Each of the five triangles, such as \( \triangle APB\), is an isosceles triangle with \( P \) being the apex and the base being one of the pentagon's sides. 3. **Perpendicular Bisectors:** - The line segment \( PM \) is perpendicular to the side \( DC \), indicating that \(M\) is not only the midpoint of \(DC\) but also on the axis of symmetry for the triangle \(PDC\). **Educational Insights:** - **Symmetry and Congruence:** The pentagon in this image is a great example to study symmetry and congruence, with point \(P\) acting as a central point from which equal paths radiate to all vertices. - **Central Angles:** Since \(P\) is the center, the angle subtended by any side of the pentagon at \(P\) will be equal, e.g., \(\angle APB\). - **Real-World Applications:** Understanding the properties of regular polygons can be helpful in various fields, including architecture, design, and even nature studies (e.g., pentagonal tiling and symmetry in flowers). By de
**Understanding the Geometry of an Octagon** In this diagram, we explore the properties of a regular octagon. A regular octagon is a polygon with eight equal sides and eight equal angles. ### Key Elements: 1. **Octagon Structure**: - The figure is a regular octagon, which means all sides and internal angles are equal. - Each side in this octagon is 9 cm in length, as indicated on one of the sides. 2. **Internal Points and Lines**: - Point \( X \) is the center of the octagon. - Vertices \( W \), \( Y \), and \( Z \) are three consecutive vertices of the octagon. - Point \( Z \) lies on the line segment connecting \( W \) and \( Y \). 3. **Inner Triangles and Right Angles**: - The line segment \( XZ \) is drawn from the center \( X \) to the midpoint \( Z \) of the base \( WY \), forming a right angle with \( WY \). - The lines \( XW \) and \( XY \) are dashed, indicating that they bisect each internal angle of the octagon, intersecting at the central point \( X \). - A small right angle is marked in red at \( ZY \), indicating that the line \( XZ \) is perpendicular to \( WY \). ### Analyzing the Triangles: - The triangle \( WXZ \) and \( YXZ \) are right triangles sharing the same height \( XZ \) from the center of the octagon to the midpoint of one of its sides. - These triangles help in calculating various properties of the octagon, such as its area, using trigonometric relationships or the Pythagorean theorem. By understanding this diagram, students can grasp the geometric properties and the symmetrical nature of regular polygons such as octagons. This foundational knowledge is crucial for more advanced geometric applications and problem-solving.
Transcribed Image Text:**Understanding the Geometry of an Octagon** In this diagram, we explore the properties of a regular octagon. A regular octagon is a polygon with eight equal sides and eight equal angles. ### Key Elements: 1. **Octagon Structure**: - The figure is a regular octagon, which means all sides and internal angles are equal. - Each side in this octagon is 9 cm in length, as indicated on one of the sides. 2. **Internal Points and Lines**: - Point \( X \) is the center of the octagon. - Vertices \( W \), \( Y \), and \( Z \) are three consecutive vertices of the octagon. - Point \( Z \) lies on the line segment connecting \( W \) and \( Y \). 3. **Inner Triangles and Right Angles**: - The line segment \( XZ \) is drawn from the center \( X \) to the midpoint \( Z \) of the base \( WY \), forming a right angle with \( WY \). - The lines \( XW \) and \( XY \) are dashed, indicating that they bisect each internal angle of the octagon, intersecting at the central point \( X \). - A small right angle is marked in red at \( ZY \), indicating that the line \( XZ \) is perpendicular to \( WY \). ### Analyzing the Triangles: - The triangle \( WXZ \) and \( YXZ \) are right triangles sharing the same height \( XZ \) from the center of the octagon to the midpoint of one of its sides. - These triangles help in calculating various properties of the octagon, such as its area, using trigonometric relationships or the Pythagorean theorem. By understanding this diagram, students can grasp the geometric properties and the symmetrical nature of regular polygons such as octagons. This foundational knowledge is crucial for more advanced geometric applications and problem-solving.
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