Midterms - Updated.pdf

MIDTERMS KARL SIMON S. CHUA, CCPE COLLEGE OF ENGINEERING AND ARCHITECTURE TOPICS COVERED IN THIS POWERPOINT ◼ Quadrilaterals and Circles ◼ Prisms and Cylinders ◼ Cones and Pyramids QUADRILATERALS QUADRILATERALS ◼ A.k.a., tetragon; quadrangle ◼ Four sided polygon ◼ Common property: The sum of t...

MIDTERMS KARL SIMON S. CHUA, CCPE COLLEGE OF ENGINEERING AND ARCHITECTURE TOPICS COVERED IN THIS POWERPOINT ◼ Quadrilaterals and Circles ◼ Prisms and Cylinders ◼ Cones and Pyramids QUADRILATERALS QUADRILATERALS ◼ A.k.a., tetragon; quadrangle ◼ Four sided polygon ◼ Common property: The sum of the interior angles is always 360o. PARTS OF QUADRILATERALS ◼ Sides: Segments joining two adjacent vertices (corners) ◼ Interior angles: Angle formed between two adjacent sides ◼ Height or Altitude: The distance between two parallel sides of a quadrilateral PARTS OF QUADRILATERALS (CONT.) ◼ Base: The bottom side that is perpendicular to the altitude ◼ Diagonal: The line segment that joins two non-adjacent vertices CLASSIFICATION OF QUADRILATERALS CLASSIFICATION OF QUADRILATERALS (CONT.) ◼ Parallelogram ◼ Two pairs of parallel sides ◼ Opposite sides and angles are congruent ◼ Adjacent/consecutive angles are supplementary ◼ Two diagonals bisect each other CLASSIFICATION OF QUADRILATERALS (CONT.) ◼ Rhombus (i.e., Diamond) ◼ All sides are equal ◼ “Equilateral parallelogram” ◼ Differs to the square in a way that only opposite angles of it are equal and its adjacent angles are not equal ◼ When the diagonals bisect each other, the resulting angle is 90o CLASSIFICATION OF QUADRILATERALS (CONT.) ◼ Rectangle ◼ All angles are right angles ◼ Square ◼ Special type of rectangle ◼ All sides are equal and all angles are the same CLASSIFICATION OF QUADRILATERALS (CONT.) ◼ Trapezoid ◼ Only a pair of parallel sides ◼ Can be right, isosceles, or scalene (trapezium) ◼ Kite ◼ Two pairs of congruent and adjacent sides ◼ Special case of rhombus ADDITIONAL PICTURES FOR REFERENCE FORMULAS: PARALLELOGRAM ◼ FORMULAS: RHOMBUS ◼ FORMULAS: RHOMBUS (CONT.) ◼ FORMULAS: RECTANGLE AND SQUARE ◼ FORMULAS: RECTANGLE AND SQUARE (FINE..) ◼ FORMULAS: KITE ◼ FORMULAS: TRAPEZOID ◼ FORMULAS: TRAPEZIUM ◼ CIRCLES CIRCLES ◼ A set of points that are equidistant from a fixed point called the center ◼ A closed two-dimensional figure that is perfectly symmetrical especially if a line is passed through the center PARTS OF A CIRCLE ◼ Circumference ◼ The length of the outer boundary of the circle ◼ The perimeter of a circle ◼ The “circle-ness” of a circle PARTS OF A CIRCLE (CONT.) ◼ Radius ◼ The line that joins the center of the circle to any points on the circle ◼ One-half of the diameter ◼ Diameter ◼ Twice the length of the radius ◼ Longest chord of the circle PARTS OF A CIRCLE (CONT.) ◼ Sector ◼ The figure formed by two radii and an included arc ◼ Chord ◼ A line segment that joins two points on the circle ◼ Segment of a secant line 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?