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Analytic Geometry Problems for Civil Engineering Board Exams in the Philippines, Lecture notes of Analytical Geometry

Technological University of the Philippines (TUP)Analytical Geometry

Understand Analytical Geometry

Typology: Lecture notes

2019/2020

Uploaded on 05/11/2023

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CIVIL ENGINEERING BOARD EXAMS PROBLEMS PHILIPPINES โ€“ OCTOBER 19, 2020
ANALYTIC GEOMETRY - known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields
of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions.
Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be
explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes'
numerical definitions and representations.
LINES โ€“ these are single dimension figure which indicates the length and determined by two points in a Cartesian coordinate plane.
CARTESIAN COORDINATE SYSTEM
DISTANCE BETWEEN TWO POINTS
EQUATIONS OF A LINE
๐ด๐‘ฅ +๐ต๐‘ฆ + ๐ถ = 0 โ†’ ๐บ๐‘’๐‘›๐‘’๐‘Ÿ๐‘Ž๐‘™ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘ก๐‘–๐‘œ๐‘›
๐‘ฆ โˆ’ ๐‘ฆ1= ๐‘š(๐‘ฅ โˆ’ ๐‘ฅ1)โ†’ ๐‘ƒ๐‘œ๐‘–๐‘›๐‘ก ๐‘ ๐‘™๐‘œ๐‘๐‘’ ๐‘“๐‘œ๐‘Ÿ๐‘š ; ๐‘š = ๐‘ ๐‘™๐‘œ๐‘๐‘’ ๐‘œ๐‘“ ๐‘กโ„Ž๐‘’ ๐‘๐‘ข๐‘Ÿ๐‘ฃ๐‘’
๐‘ฅ
๐‘Ž+๐‘ฆ
๐‘= 1 โ†’ ๐ผ๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘’๐‘๐‘ก ๐‘“๐‘œ๐‘Ÿ๐‘š ; ๐‘Ž = ๐‘ฅ โˆ’ ๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘’๐‘๐‘ก ๐‘Ž๐‘›๐‘‘ ๐‘ = ๐‘ฆ โˆ’ ๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘’๐‘๐‘ก
๐‘ฆ โˆ’ ๐‘ฆ1
๐‘ฅ โˆ’ ๐‘ฅ1=๐‘ฆ2โˆ’ ๐‘ฆ1
๐‘ฅ2โˆ’ ๐‘ฅ1โ†’ ๐‘‡๐‘ค๐‘œ ๐‘๐‘œ๐‘–๐‘›๐‘ก ๐‘“๐‘œ๐‘Ÿ๐‘š
๐ด๐‘ฅ +๐ต๐‘ฆ = ๐ถ โ†’ ๐‘†๐‘ก๐‘Ž๐‘›๐‘‘๐‘Ž๐‘Ÿ๐‘‘ ๐น๐‘œ๐‘Ÿ๐‘š
๐‘ = ๐ด cos๐œƒ + ๐ต sin๐œƒ โ†’ ๐‘ƒ๐‘œ๐‘™๐‘Ž๐‘Ÿ ๐‘œ๐‘Ÿ ๐‘๐‘œ๐‘Ÿ๐‘š๐‘Ž๐‘™ ๐น๐‘œ๐‘Ÿ๐‘š
SLOPE OF THE LINE MIDPOINT OF A SEGMENT
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CIVIL ENGINEERING BOARD EXAMS PROBLEMS PHILIPPINES โ€“ OCTOBER 19, 2020

ANALYTIC GEOMETRY - known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. LINES โ€“ these are single dimension figure which indicates the length and determined by two points in a Cartesian coordinate plane. CARTESIAN COORDINATE SYSTEM DISTANCE BETWEEN TWO POINTS EQUATIONS OF A LINE ๐ด๐‘ฅ + ๐ต๐‘ฆ + ๐ถ = 0 โ†’ ๐บ๐‘’๐‘›๐‘’๐‘Ÿ๐‘Ž๐‘™ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐‘ฆ โˆ’ ๐‘ฆ 1 = ๐‘š(๐‘ฅ โˆ’ ๐‘ฅ 1 ) โ†’ ๐‘ƒ๐‘œ๐‘–๐‘›๐‘ก ๐‘ ๐‘™๐‘œ๐‘๐‘’ ๐‘“๐‘œ๐‘Ÿ๐‘š ; ๐‘š = ๐‘ ๐‘™๐‘œ๐‘๐‘’ ๐‘œ๐‘“ ๐‘กโ„Ž๐‘’ ๐‘๐‘ข๐‘Ÿ๐‘ฃ๐‘’ ๐‘ฅ ๐‘Ž

๐‘ = ๐ด cos ๐œƒ + ๐ต sin ๐œƒ โ†’ ๐‘ƒ๐‘œ๐‘™๐‘Ž๐‘Ÿ ๐‘œ๐‘Ÿ ๐‘๐‘œ๐‘Ÿ๐‘š๐‘Ž๐‘™ ๐น๐‘œ๐‘Ÿ๐‘š SLOPE OF THE LINE MIDPOINT OF A SEGMENT

DIVISION OF LINE SEGMENTS

DISTANCE FROM A POINT= TO A LINE.

DISTANCE BETWEET TWO PARALLEL LINES: RELATIONSHIP BETWEEN PERPENDICULAR AND PARALEL LINES:

ANGLE BETWEEN TWO LINES: AREA OF THE POLYGON USING COORDINATES (Shoelace Method): Note: For the slope and the angles between curves, see Differential Calculus.

  1. A point divides internally line segment joining (8,9) and (-7,4) in the ratio 2:3. Find the coordinates. SOLUTION:
  2. Find the distance between P (4,5) to a line 8x + 5y = 20 SOLUTION:
  3. Find the angle between the slopes of 1/2 and 2. SOLUTION:
  4. Determine the kind of a triangle given the points A (1,3) , B (-1, 2) and C (5,3). SOLUTION:
  1. Assume the graph has 1 unit each grid, determine the line of the graph shown. SOLUTION:
  2. Find the area of the quadrilateral given the points (5,2) , (4,3) , (2,4) and (-8 , - 1). Using the formula: ๐ด =

[

๐‘ฆ 1 ]

[^5

]

EXERCISES โ€“ Answer the following questions.

  1. Find line in standard form given that P (0 ,9) and m = - 2. Ans. 2x + y = 9
  2. Find the area covered given the points (1,4) , (7,0) , (5 , - 3) and (-1 , 1). Ans. 26
  3. Find the equation of the line which cuts off an intercept 3 on the positive direction of x-axis and an intercept 5 on the negative direction of y-axis. Ans. 5x โ€“ 3y = 15
  4. A and B are the vertices of the base of an isosceles triangle ABC. A is at point (1,2) and B at (4,1). Compute the area of the triangle given the ordinate of 4. Ans. 10.
  5. Find the acute angle between the two lines 2x + 4y โ€“ 5 = 0 and 7x โ€“ 3y + 2 = 0. Ans. 52.594ยฐ
  6. Find the equation of the line parallel to 4x + 3y = 12 and passing through (โˆ’12, 4) Ans. 4x +3y + 36 = 0
  7. Calculate the shortest distance between the parallel line y = 1/2 x and y = 1/2 x โ€“ 5. Ans. 4.
  8. Write the equation of the line in slope intercept form if the line passes through (3,0) and perpendicular to the line y = 3x +1. Ans. y = - 1/3 x + 3
  9. (ECE Board)
  10. Determine the coordinates of the point which is three fifths of the way from the point (2, - 5) to the point (-3 , 5). Ans. (-1,1)
  11. If the coordinates of ๐ด and ๐ต are (5, 5) and (โˆ’1, โˆ’4) respectively, find the coordinates of the point ๐ถ that divides ๐ด๐ต internally in the ratio 2 : 1. Ans. (1, - 1)
  12. Find the distance from the point (-6,8) to the line y = - 3x +10. Ans. 6.
  13. If M (1, 1) is the midpoint of the line segment joining A (3, 1) and B (x, y), find the coordinates of B. Ans. (5,3)
  14. Find the distance between two lines 5x + 3y + 6 = 0 and 5x + 3y โ€“ 6 = 0. Ans. 12/โˆš
  15. Given the vertices A (1,4) , B(3,6) , C(6,3) and D(4,1) , determine the perimeter. Ans. 14.
  16. Find the equation of line through point (3,2) and making angle 45ยฐ with the line x-2y = 3. Ans. 3x โ€“ y โ€“ 0 and x + 3y โ€“ 9 = 0.
  17. Determine the distance between the points (1.1 , 2.2) and (3.3 , 4.4). Ans. 3.
  18. Consider the points (-1,2) and (2, - 3) , find the point which divides AB internally in the ratio 3:1. Ans. (5/4, - 7/4)
  19. Without graphing, find the gradient of the equation 3y = - 6x +2. Ans. - 2
  20. Find the point whose coordinates are equal and equidistant from the points (-2 , 3) and (1,4). Ans. (-0.5 , 0.5)
  1. If the given circles are tangent to each other, the radical axis is tangent to each other. The radical axis is always perpendicular to the line joining the centers of the given circles.
  2. If the given circles have no common point, the radical axis is between the given circles. The radical axis is always perpendicular to the line joining the centers of a given circles. Note: Recall the non linear systems of equations. EXAMPLES:
  3. (CE Board) A circle has the equation x^2 + y^2 โ€“ 6x + 12y + 9 = 0. Find the radius of the circle. SOLUTION:
  4. Rewrite in standard form: x^2 + y^2 + 4x โ€“ 6y โ€“ 23 = 0. SOLUTION:
  1. Given the circles x^2 + y^2 โ€“ 4x โ€“ 6y = 3 and x^2 + y^2 + 6x +4y โ€“ 7 = 0 , find their intersections.
  2. Find the equation of the circle that passes through the points (1 , - 6) , (2,1) and (5,2).

REFERENCES

  1. Analytic Geometry by Rainville
  2. Engineering Mathematics by Gillesania
  3. Engineering Mathematics by Bird
  4. https://www.chilimath.com/lessons/intermediate-algebra/distance-formula/
  5. Intermediate Algebra by Elayn Martin Guy
  6. https://www.math-only-math.com/division-of-line-segment.html
  7. https://www.tpub.com/math2/5.htm
  8. Glencoeโ€™s Advanced Mathematical Concepts
  9. Scahumโ€™s Outlines of Precalculus
  10. https://calcworkshop.com/graphing-linear-equations/point-slope-form/
  11. https://www.exeter.k12.pa.us/cms/lib6/PA01000700/Centricity/Domain/118/Day%209%20-%20Point%20Slope%20Homework%20complete.pdf
  12. https://www.math-only-math.com/straight-line-in-intercept-form.html
  13. Engineering Mathematics Vol. 1 by Besavilla
  14. Analytic Geometry by AC Burdette
  15. MMC Questionnaires
  16. https://www.ck12.org/geometry/Distance-Between-Parallel-Lines/lesson/Distance-Between-Parallel-Lines-GEOM/
  17. Algebra and Trigonometry by Cynthia Young
  18. Engineering Mathematics by Excel Review Center
  19. https://www.scevmath.org/uploads/2/8/9/4/28940543/3.4_distance_from_a_point_to_a_line.pdf
  20. College Algebra with Trigonometry by Barnett
  21. https://www.toppr.com/guides/maths/straight-lines/distance-of-point-from-a-line/
  22. https://www.engageny.org/sites/default/files/downloadable-resources/2014/Aug/geometry-m4-topic-c-lesson- 10 - teacher.pdf
  23. https://www.mathstopia.net/coordinate-geometry/angle-two-lines
  24. https://www.cuemath.com/geometry/coordinate-geometry-internal-division/
  25. https://www.math-only-math.com/circle-passing-through-three-given-points.html
  26. http://www.nabla.hr/PC-CircleAndLineCnt3.htm
  27. Introductory and Intermediate Algebra by Blitzer
  28. Euclid Math Contest Questionnaires
  29. https://www.superprof.co.uk/resources/academic/maths/analytical-geometry/conics/equation-of-a-circle-problems.html#chapter_exercise- 12
  30. 1,001 Geometry Problems
  31. https://tutorial.math.lamar.edu/Solutions/Alg/Circles/Prob6.aspx
  32. https://cdn.kutasoftware.com/Worksheets/Geo/11-Equations%20of%20Circles.pdf
  33. https://mathforums.com/threads/difficult-question-equation-of-a-circle.180/
  34. https://www.mathwarehouse.com/sheets/algebra/equation-of-circle-worksheet.php