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UNIT 9 GEOMETRY 1.- Triangle: A triangle is a figure formed when three non collinear points are connected by segments. Each pair of segments forms an angle of the triangle. The vertex of each angle is a vertex of the triangle. The sum of the measures of the angles of a triangle is 180. The sum of the lengths of any two sides of a triangle must be greater than the third side. That is: a+b>c a+c>b b+c>a The subtraction of the lengths of any two sides of a triangle must be smaller than the third side. Triangles can be classified by: Their sides: Equilateral Isosceles Scalene All three sides have equal lengths Exactly two equal sides All sides have different lengths Their angles: Acute Right Obtuse All interior angles are acute (<90º) One angle is a right angle (90º) One angle is obtuse (>90º) Area of a triangle There are several ways to compute the area of a triangle: 1. When you know the lenght of the base and the height, you can use the formula: 2.- Square: A square is a parallelogram with 4 congruent sides and 4 right angles. A square is a particular case of a rectangle and a rhombus simultaneously. So, it shows both the properties of rhombus and rectangle simultaneously. Square can be differentiated from a rectangle and rhombus due to following properties: 1. Unlike rectangle square needs to have all its sides equal. 2. Unlike rhombus square needs to have all angles equal to 90 degree. Area of a square If l is the side-length of a square, the area of the square is: 3.- Rectangle: A rectangle is a four-sided polygon with four right angles, whose opposite sides are parallel and are equal. A rectangle is a particular sort of parallelogram, but can't say that all parallelograms would be rectangles, because a rectangle is a shape where opposites sides are parallel and all the corners are 90 degree angles. Some parallelograms would be rectangles, but not all. Area of a rectangle The area of a rectangle is the product of its width and length. 4.- Rhombus: A Rhombus is a four-sided polygon having all four sides of equal length and whose opposite sides are parallel. Each rhombus has two diagonals. The biggest one is called long diagonal and the smaller one is called short diagonal. (Note that the diagonals of a rhombus are perpendicular). Area of a rhombus The area of any rhombus is equal to one-half the product of the lengths D and d of its diagonals. 5.- Trapezoid: A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The nonparallel sides are called legs. Area of a trapezoid The area of a trapezoid is given by the formula: , where B, b are the lengths of the two bases and h is the altitude of the trapezoid. 6.- Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides. The sum of the measures of the angles of a parallelogram is 360. Squares, rectangles and rhombus are particular types of parallelograms. Area of a parallelogram The area of a parallelogram is given by the formula: 7.- Regular polygon: A regular polygon is a polygon which is equiangular (all angles are congruent) and equilateral (all sides have the same length). Attributes: • • • • • • The sides are the straight line segments that make up the polygon. The vertex is a corner of the polygon. In any polygon, the number of sides and vertices are always equal. The center is the point inside a regular polygon that is equidistant from each vertex. The apothem of a regular polygon is the line from the center to the midpoint of a side. The radius is the distance from the center to any vertex. By definition, all sides are the same length, so the perimeter is simply the length of a side times the number of sides. Examples of regular polygons: Area of a regular polygon: 8.- Circle: A circle is a plane figure, bounded by a single curve line called its circumference, every part of which is equally distant from a point within it, called the center. A circle sector is any piece of the circle between two radial lines (shaded in both dark and clear grey). A segment of a circle is the region between a chord of a circle and its associated arc (shaded in dark grey). An annulus is the region lying between two concentric circles Areas 9.- Pythagorean Theorem: Pythagorean Theorem: In a right angled triangle, with sides (legs) a and b, and hypotenuse c, then c² = a²+b². The square of the hypotenuse is the sum of the squares of the legs. A "3,4,5" triangle has a right angle in it, so the formula should work. 52=32+42 25=9+16 25=25 Yes, it works!!! Apply the Pythagorean Theorem: If you know any two of the sides in a right triangle, you can find the dimensions for the missing side using the Pythagorean Theorem. a) Using the Pythagorean Theorem to find a missing leg We can use the Pythagorean theorem to find a missing leg of a triangle, but only if we know the length measure of the hypotenuse and the other one of the legs. Find x 52=32+x2 x2=52-32 x2=25-9 x2=16 x=4 b) Using the Pythagorean Theorem to find the hypotenuse We can use the Pythagorean theorem to find the hypotenuse of a right angled triangle, but only if we know the length measure of the two legs. Find x c2=a2+b2 c2=82+122 c2=64+144 c2=208