- 1. CHAPTER 9 PLANE FIGURES Triangle A triangle is one of the basic shapes in geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted . In Euclidean geometry any three points, when non- collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). All triangles have 3 sides and 3 angles which always add up to 180°. The Triangle Inequality Theorem states that: Any side of a triangle must be: less than the sum of the other 2 sides and greater than the difference of the other 2 sides. CLASSIFICATION OF TRAINGLES Triangles are classified in 2 ways- 1) By the number of equal sides they have: • scalene - all 3 sides have different lengths • isosceles - 2 sides have equal lengths • equilateral - all 3 sides are equal 2) By the types of angles they have: • acute triangle - all 3 angles are acute (less than 90°) • right triangle - has one right angle (a right angle = 90°) • obtuse triangle - has one obtuse angle (an obtuse angle is greater than 90° and less than 180°). Relationship Between Sides of a Triangle Here is a typical question: Two sides of a triangle measure 2cm and 5cm. What is a possible measurement of the third side? Sometimes you'll be given multiple choice, and sometimes you'll have to represent the possible range of answers. Look at the diagram at left. In the upper picture, I've arranged the two sides so that they are separated quite far apart. Notice that the missing side cannot possibly be larger than 7, which is the sum of the two sides. If it was, then it would not be able to connect to the other two sides. It can't be equal to 7 either. If it was, then the two other sides would have to be lying flat along a straight line, and of course we then wouldn't have a triangle. They have to be angled at least slightly. This means that x < 7. Now let's look at the lower picture. I've arranged the sides so that they are quite close together, forming a small angle. Notice that x must be greater than 3, which is the difference of the two sides. If it was less than 3, it would be too short to connect. If it was exactly 3, then the sides of 2 and 5 would actually be on top of each other, and that wouldn't be a triangle anymore. The sides of 2 and 5 must be angled at least slightly. This means that x > 3. We can say that 3 < x < 7. This means that x must be between 3 and 7, but not equal to either. Any answer within this range will be correct. Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or edges) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6- sided) and so on. The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side." Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is This is a special case of the n-gon interior angle sum formula (n − 2) × 180°. In a crossed quadrilateral, the four interior angles on either side of the crossing add up to 720°.[1] All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges. Convex polygons A convex polygon is a simple polygon whose interior is a convex set.[1] The following properties of a simple polygon are both equivalent to convexity: Every internal angle is less than or equal to 180 degrees. Every line segment between two vertices remains inside or on the boundary of the polygon. A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees. Equivalently, a polygon is strictly convex if every line segment between two nonadjacent vertices of the polygon is strictly interior to the polygon except at its endpoints. Every nondegenerate triangle is strictly convex. Circle A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point. A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk. A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant. A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0.
- 2. Arcs and Chords In Figure 1, circle O has radii OA, OB, OC and OD If chords AB and CD are of equal length, it can be shown that Δ AOB ≅ Δ DOC. This would make m ∠1 = m ∠2, which in turn would make m = m . This is stated as a theorem. Figure 1 A circle with four radii and two chords drawn. Theorem 78: In a circle, if two chords are equal in measure, then their corresponding minor arcs are equal in measure. The converse of this theorem is also true. Theorem 79: In a circle, if two minor arcs are equal in measure, then their corresponding chords are equal in measure. Example 1: Use Figure 2 to determine the following. (a) If AB = CD, and = 60°, find m CD. (b) If m = and EF = 8, find GH. Figure 2 The relationship between equality of the measures of (nondiameter) chords and equality of the measures of their corresponding minor arcs. m = 60° (Theorem 78) GH = 8 (Theorem 79) Some additional theorems about chords in a circle are presented below without explanation. These theorems can be used to solve many types of problems. Theorem 80: If a diameter is perpendicular to a chord, then it bisects the chord and its arcs. In Figure 3, UT, diameter QS is perpendicular to chord QS By Theorem 80, QR = RS, m = m , and m = m . Figure 3 A diameter that is perpendicular to a chord. Theorem 81: In a circle, if two chords are equal in measure, then they are equidistant from the center. In Figure 4, if AB = CD, then by Theorem 81, OX = OY. Figure 4 In a circle, the relationship between two chords being equal in measure and being equidistant from the center. Theorem 82: In a circle, if two chords are equidistant from the center of a circle, then the two chords are equal in measure. In Figure 5, if OX = OY, then by Theorem 82, AB = CD. Example 2: Use Figure to find x. Figure 5 A circle with two minor arcs equal in measure. Example 3: Use Figure 6, in which m = 115°, m = 115°, and BD = 10, to find AC. Figure 6 A circle with two minor arcs equal in measure. Example 4: Use Figure 7, in which AB = 10, OA = 13, and m ∠ AOB = 55°, to find OM, m and m . Figure 7 A circle with a diameter perpendicular to a chord. So, ST ⊥ AB, and ST is a diameter. Theorem 80 says that AM = BM. Since AB = 10, then AM = 5. Now consider right triangle AMO. Since OA = 13 and AM = 5, OM can be found by using the Pythagorean Theorem.