Part 02 - Midterms - Updated (2) PDF
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College of Engineering and Architecture
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Summary
This document provides notes on various geometrical figures including circles, parabolas, and ellipses, along with their characteristics, formulas, and applications. Topics such as Simpson's rule for calculating areas of irregular figures and examples using measurements are also included.
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PARTS OF A CIRCLE (CONT.) ◼ Segment ◼ The region bounded by a chord of a circle and the intercepted arc of the circle ◼ Arc ◼ A portion of the circle that contains two endpoints and all the points of the circle between those endpoints PARTS OF A CIRCLE (CONT.) ◼ Tangent line ◼ This l...
PARTS OF A CIRCLE (CONT.) ◼ Segment ◼ The region bounded by a chord of a circle and the intercepted arc of the circle ◼ Arc ◼ A portion of the circle that contains two endpoints and all the points of the circle between those endpoints PARTS OF A CIRCLE (CONT.) ◼ Tangent line ◼ This line intersects that circle at exactly one point (i.e., point of tangency) ◼ It doesn’t cross the circle. It simply “touches” it ◼ Secant line ◼ A line that intersects a circle at two points CHARACTERISTICS OF A CIRCLE ◼ A chord divides a circle into two regions, the major segment (larger part) and the minor segment (smaller part) ◼ The diameter is the longest chord ◼ Circles are congruent if they have equal radii. However, the still remain similar even if their radii are not ◼ Circles can circumscribe a rectangle, trapezium, triangle, square, and kite CHARACTERISTICS OF A CIRCLE (CONT.) ◼ The interior of the circle is the set of all points within the boundary of the circle whose distances from the center are always less than the radius ◼ The exterior of the circle is the set of all points outside of the boundary of the circle and whose distances from the center are always greater than the radius ◼ The distance from the center of the circle to the longest chord is zero ◼ The perpendicular distance from the center of the circle decreases when the length of the chord increases CHARACTERISTICS OF A CIRCLE (CONT.) ◼ The central angle is the angle whose vertex lies at the center of the circle and the two sides are the two radii ◼ The inscribed angle is the angle whose vertex lies on the circle and whose two sides are chords of the circle CHARACTERISTICS OF A CIRCLE (CONT.) ◼ Every tangent line of a circle is perpendicular to the radius of the circle drawn through the point of tangency ◼ Central angles of similar circles have the same ratio as their intercepted arcs ◼ You can create an isosceles triangle by joining two ends of a chord to two radii of a circle CIRCLE THEOREMS (CONT.) ◼ Lines of Centers of Tangent Circles ◼ The line of centers of two tangent circles passes through the point of tangency ◼ Distance of circles (left side) is equal to r1+ r2 ◼ Distance of circles (right side) is equal to r2 – r1 FORMULAS ◼ FORMULAS (CONT.) ◼ AREA FORMULAS ◼ MISCELLANEOUS AREA FORMULAS ◼ EQUILATERAL TRIANGLE INSCRIBED IN A CIRCLE ◼ EQUILATERAL TRIANGLE CIRCUMSCRIBING A CIRCLE ◼ MISCELLANEOUS PLANE FIGURES STAR ◼ Figure which generally consists of a polygon with triangles on its sides ◼ Regular star (i.e., regular star, German star, or witch star) ◼ Hexagram (i.e., David’s star, Solomon’s star) STAR (CONT.) ◼ Area ◼ Solve for the area of the triangles first ◼ Afterward, solve the area of the remaining polygon ◼ Finally, find the sum of the areas ELLIPSE ◼ A plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant ◼ Eccentricity is less than 1 ◼ The circle is a special type of ellipse where the focal points are equal ◼ TL;DR – Oblong ELLIPSE (CONT.) ◼ PARABOLIC SECTION ◼ A plane curve which is mirror-symmetrical and is approximately U-shaped ◼ Eccentricity is equal to 1 ◼ Spandrel is the almost triangular space between one side of the outer curve of an arch ◼ The space between the shoulders of adjoining arches and the ceiling or molding above. PARABOLIC SECTION (CONT.) ◼ COMPOSITE PLANE FIGURES ◼ These are figures that are made up of different geometrical figures ◼ To solve for its area, we “break down” the figure (i.e., do the job by parts) SIMPSON’S RULE ◼ EXAMPLE ◼ Find the area of the figure that has the following measurements: ◼ y0 = 5.2, y1 = 3.2, y2 = 6.1, y3 = 5.4 ◼ y4 = 6.2, y5 = 4.6, y6 = 5.3, y7 = 3.6 ◼ y8 = 6.7 ◼ Interval of 2 units QE20 (ANSWER) ANY QUESTIONS?
