Geometric Analysis I (Lines Planes Triangles and Polygons)

Question

Geometry deals with measures and properties of points, lines, planes, surfaces and solid. We would go into detail the following as well as their respective properties: Lines Angles Polygons Triangles Lines A line in geometry is a straight one-dimensional figure that has no thickness and extends infinitely in both directions. It is typically represented with two arrowheads and named using any two points on it, like line AB, or simply denoted as ℓ. Properties of a line It has length, but no width or height. It is made up of an infinite number of points. It extends endlessly in both directions. Angles An angle is formed when two straight lines (or rays) meet at a common point called the vertex. It measures the amount of turn or rotation between the two lines around the vertex. Properties of Angles The two lines forming the angle are called arms or sides. Angles are measured in degrees (°). A full circle is 360°, a right angle is 90°, and a straight angle is 180°. Types of angles: Acute angle: less than 90° Acute angle lie between 0o and less than 90o. ∠AOB is an Acute Angle because 0o < Θ < 90o. Right angle: A right angle is an angle that measures exactly 90 degrees (90°). It is formed when two lines or rays meet to create a perfect L-shape, like the corners of a square or rectangle. Symbol: ⊾ . It divides a straight angle (180°) into two equal parts. It’s the standard for perpendicular lines ∠AOB is a Right Angle because Θ = 90o. Obtuse angle An Obtus angle is an angle greater than 90° but less than 180° ∠AOB is an Obtus Angle because 90o < Θ < 180o. It's wider than a right anglen and still opens less than a straight line (180°). Example: 120°. Straight angle: A straight angle is an angle that measures exactly 180 degrees (180°). It forms a straight line. The two arms of the angle point in exactly opposite directions. It looks like a flat line, but it's still considered an angle because it's formed by two rays meeting at a point. ∠AOB is a Straight Line Angle because Θ = 180o. Reflex angle: A reflex angle is an angle that measures more than 180° but less than 360°. ∠AOB is a Reflex Angle because 180o < Θ < 360o. Full angle: exactly 360° Angle Relationships Angles can relate to each other in various ways based on their positions. An understanding of these relationships can enable us to solve geometric problems. (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); Some relationship that exists include: (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); Complementary Angles Complementary Angles are two angles whose sum is exactly 90 degrees (90°). Each angle is said to complement the other, i.e θ1 + θ2 = 90o. Example is 30o and 60o, the sume of these angles give 90o. Supplementary Angles Supplementary Angles are two angles whose sum is exactly 180 degrees (180°). When two angles are supplementary, together they form a straight line i.e θ1 + θ2 = 180o. Vertically Opposite Angles Vertically Opposite Angles are the angles formed opposite each other when two lines intersect From the above image, ∠AOB = ∠COD and ∠AOC = ∠BOD. Adjacent Angles Adjacent Angles are two angles that share a common side and a common vertex, but do not overlap. ∠AOC and ∠BOC are adjacent angles as they both have a common side which is OC. Angles on one side of a line Angles on one side of a straight line that add up to 180° are called supplementary angles. When these angles are adjacent and lie on a straight line, they are also known as a linear pair. Polygons A polygon is a 2-dimensional closed shape made up of straight line segments connected end to end having n number of sides. If all the sides and the angles of this polygon are equivalent then the polygon is considered to be a regular polygon. The word polygon came from the greek word poly meaning many and gonia meaning angle. The most common type of polygon is the regular polygon which is a convex polygon with equal side lengths and angles. Properties of a regular polygon Every internal angle is at most 180 degrees. Interior Angle + Exterior Angle = 180 degrees. The number of diagonals in an n-sided polygon = n(n-3)/2. The sum total of all the exterior angles of any polygon is 360o. The measure of each exterior angle of a regular polygon = 360o/n. The ratio of the sides of a polygon to the diagonals of a polygon is 2:(n-3). The ratio of the interior angles of a regular polygon to its exterior angle is (n-2):2. The sum total of all the interior angles of any polygon = (2n - 4) * 90o. So each interior angle = (2n-4)n*90 Every line segment between the two vertices of the polygon does not go outside the polygon. In other words, all the diagonals of the polygon remain inside its boundary. Every triangle is strictly a convex polygon. Types of polygon according to sides: Number of sides Name of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon Area and Perimeter of a regular polygon For a regular polygon : Perimeter (P) $ P = n \times s $ $n$ = number of sides $s$ = length of one side Area (A) $ A = \frac{1}{2} \times P \times a = \frac{1}{2} \times n \times s \times a $ a = apothem (distance from center to midpoint of a side) $P = n \times s$ (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); Example 1 n = 6, s = 4, a = 3.5 P = 6 * 4 = 24 units $A = \frac{1}{2} \times 24 \times 3.5 = 42 \ \text{units}^2$ (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); Example 2 If PQRSTUVW is a regular octagon, then what is the measure in degrees of ∠WPQ - ∠PVQ? ∠WPQ = Interior angle = (2*8-4)8*90= 135o. In △WVP, ∠WVP = ∠WPV And ∠VWP = 135o. So, ∠WVP = 45/2 = 22.5o. Hence, ∠WPQ - ∠PVQ = 112.5o. Example 3 The exterior angle of a regular polygon is 1/3rd of its interior angle. How many sides does the polygon have? Assume that the value of the exterior angle = P, then the value of the interior = 3P. So that P + 3P = 180o. P = 45o (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); The number of sides of the polygone = 360/45 = 8 sides. (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); Triangles A triangle is a figure enclosed by three sides. In the figure given below, ABC is a triangle with sides AB, BC and CA measuring c, a and b units respectively. Line AD represents the height of the triangle corresponding to the side BC and denoted by h In any triangle, Area = 12* BC * AD = 12*a*h Properties of a triangle The sum of all the angle in a triangle is 180o. The sum of lengths of the two sides > length of the third side. The difference of any two sides of any triangle < length of the third side. The area of any triangle can be found by several methods: Area of any triangle = 1/2 x base x perpendicular height. Area of any triangle = S(S-a)(S-b)(S-c) Where S is the semi-perimeter of the triangle and a, b and c are the sides of the triangle. Area of any triabgle = 1/2 x bc x Sin A Example 4 What is the number of distinct triangles with integral valued sides and perimeter as 14? a) 6 b) 5 c) 4 d) 3 The sum of the length of two sides > the length of the third side. So the maximum length of any particular side can be 6 units Now if a = 6 then b + c = 8, so the possible sets (6, 6, 2), (6, 4, 4), (6, 5, 3), (5, 5, 4). Thus the number of distinct triangles = 4. Classification of triangles A. Based upon sides B. Based upon angles (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); A.Based upon side 1. Scalene Triangle All sides are of a different length. Area = S(S-a)(S-b)(S-c) S(semi perimeter) = a+b+c2 (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); 2. Isosceles triangle A triangle whose two sides are of equal length. Height = 4a2-b22 Area = b44a2-b2 3. Equilateral triangle A triangle in which all sides are of equal lengths. Height = 32a, Area = 32a2 where a is any side (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); B. Based upon angles 1. Right angled triangle A triangle in which one of the 3 angles is 90o. The side opposite this 90o angle is called the hypotenuse which is usually the longest side. Area = 12*base*height (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({});