Module 7 MATH 111 Final COPY Geometry Second COPY Editedchapter 888 PDF

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Math Geometry, this can help you if you don't easily understand you current material....

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MODULE 7 : Mathematical system in Geometry After studying this module, you should be able to: 1. 2. 3’ 4. 5. 6.

Identify and define points, lines, line segments, rays, and planes. Classify angles as acute, right, obtuse, or straight nderstand geometrical terminology for angles, triangles, quadrilaterals and circles Use geometrical results to determine unknown angles Recognise line and rotational symmetries Find the areas of triangles, quadrilaterals and circles and shapes based on these.

TOPICS/ CONTENT 1. Basic Geometric Concepts and Figures 2. Perimeter, Circumference and Area 3. Quadrilaterals; Volume of Geometric Solids 4 Points, Lines, Planes and Angles. 5. Polygons; Perimeter, areas, 6. Circles, Triangles,

DEFINITION: Geometry is the study of points, lines, angles, surfaces, and solids. Point: A point is a location in space. It is represented by a dot. Point are usually named with a upper case letter. For example, we refer to the following as "point A" A point in geometry is a location. It has no size i.e. no width, no length and no depth. A point is shown by a dot. Line: A line is a collection of points that extend forever. The following is a line. The two arrows are used to show that it extends forever. A line is defined as a line of points that extends infinitely in two directions. It has one dimension, length. Points that are on the same line are called collinear points.

A line is defined by two points and is written as shown below with an arrowhead. AB↔AB↔ Line segment: A part of a line that has defined endpoints is called a line segment. A line segment as the segment between A and B above is written as: AB¯¯¯¯¯¯¯¯AB¯ Measure line segments The length of a line segment can be measured (unlike a line) because it has two endpoints. As we have learnt previously the line segment can be written as AB¯¯¯¯¯¯¯¯AB¯ While the length or the measure is simply written AB. The length could either be determined in Metric units (e.g. millimeters, centimeters or meters) or Customary units (e.g. inches or foot). Two lines could have the same measure but still not be identical.

AB and CD have the exact same measure and are said to be congruent and is noted as AB¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯AB¯≅CD¯ This is read as the line AB is congruent to the line CD. Finding distances and midpoints If we want to find the distance between two points on a number line we use the distance formula: AB=|b−a|or|a−b| Example Point A is on the coordinate 4 and point B is on the coordinate -1. AB=|4−(−1)|=|4+1|=|5|=5 If we want to find the distance between two points in a coordinate plane we use a different formula that is based on the Pythagorean Theorem where (x1,y1) and (x2,y2) are the coordinates and d marks the distance: d = √ (x2−x1)2+(y2−y1)2

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The point that is exactly in the middle between two points is called the midpoint and is found by using one of the two following equations. Method 1: For a number line with the coordinates a and b as endpoints: midpoint= (a+b)/2 Method 2: If we are working in a coordinate plane where the endpoints has the coordinates (x1,y1) and (x2,y2) then the midpoint coordinates is found by using the following formula: midpoint=(x1+x2)/2,

(y1+y2)/2

Finding distances and midpoints If we want to find the distance between two points on a number line we use the distance formula: AB=|b−a|or|a−b|AB=|b−a|or|a−b| Example Point A is on the coordinate 4 and point B is on the coordinate -1. AB=|4−(−1)|=|4+1|=|5|=5AB=|4−(−1)|=|4+1|=|5|=5 Example Point A is on the coordinate 4 and point B is on the coordinate -1. AB=|4−(−1)|=|4+1|=|5|=5AB=|4−(−1)|=|4+1|=|5|=5

Angle: Two rays with the same endpoint is an angle. The following is an angle. Measure and classify an angle A line that has one defined endpoint is called a ray and extends endlessly in one direction. A ray is named after the endpoint and another point on the ray e.g.

AB→AB→ The angle that is formed between two rays with the same endpoint is measured in degrees. The point is called the vertex

The vertex is written as ∡CAB∡CAB In algebra we used the coordinate plane to graph and solve equations. You can plot lines, line segments, rays and angles in a coordinate plane.

In the coordinate plane above we have two rays BA→andBD→BA→andBD→ That form an angle with the vertex in point B. You can use the coordinate plane to measure the length of a line segment. Point B is at (-2, -2) and C (1. -2). The distance between the two points is 1 - (-2) = 3 units. Angles can be either straight, right, acute or obtuse.

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An angle is a fraction of a circle where the whole circle is 360°. A straight angle is the same as half the circle and is 180° whereas a right angle is a quarter of a circle and is 90°. You measure the size of an angle with a protractor.

Two angles with the same measure are called congruent angles. Congruent angles are denoted as ∠A≅∠B∠A≅∠B Or could be shown by an arc on the figure to indicate which angles that are congruent.

Two angles whose measures together are 180° are called supplementary e.g. two right angles are supplementary since 90° + 90° = 180°. Two angles whose measures together are 90° are called complementary.

m∠A+m∠B=180∘m∠A+m∠B=180∘ m∠C+m∠D=90∘ Some common angle properties The sum of angles at a point is 360˚.

Vertical angles are equal.

The sum of complementary angles is 90˚.

The sum of angles on a straight line is 180˚.

Alternate Angles (Angles found in a Z-shaped figure)

Corresponding Angles (Angles found in a F-shaped figure)

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Interior Angles (Angles found in a C-shaped or U-shaped figure) Interior angles are supplementary. Supplementary angles are angles that add up to 180˚.

The sum of angles in a triangle is 180˚.

An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

The sum of interior angles of a quadrilateral is 360˚.

-------------------Plane: A plane is a flat surface like a piece of paper. It extends in all directions. We can use arrows to show that it extends in all directions forever. The following is a plane

Parallel lines When two lines never meet in space or on a plane no matter how long we extend them, we say that they are parallel lines The following lines are parallel. Intersecting lines: When lines meet in space or on a plane, we say that they are intersecting linesThe following are intersecting lines.

Vertex: The point where two rays meet is called a vertex. In the angle above, point A is a vertex. Geometric Shapes: List, Definition, Types of Geometric Shapes Geometric Shapes can be defined as figure or area closed by a boundary which is created by combining the specific amount of curves, points, and lines. Different geometric shapes are Triangle, Circle, Square List of Geometric Shapes: Square 1. 2. Circle 3. Rectangle 4. Triangle 5. Polygons 6. Parallelogram Square A square is a four-sided figure which is created by connecting 4 line segments. The line segments in the square are all of the equal lengths and they come together to form 4 right angles.

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Circle A circle which is another shape of geometry has no straight lines. It is rather a combination of curves that are all connected. In a circle, there are no angles to be found.

Rectangle Similar to a square, a rectangle is also created by connecting four line segments. However, the only difference between a square and a rectangle is that in a rectangle, there are two line segments which are longer than the other two line segments.

So, in geometry, a rectangle is also described as an elongated square. Also, in a rectangle, the four corners come together to form four right angles. Triangle Triangle comprises three connected line segments. Unlike, a rectangle or a square, in a triangle, the angles can be of distinct measurements. They aren’t always the right angles. Triangles are named, depending upon the type of angles which is found within the triangle itself. For instance, if a triangle has one right angle, it will be known as a right-angled triangle.

However, in case all the angles of a triangle are less than 90 degrees, then it will be called as an acute-angled triangle. If any, one of the angles in the triangle measures more than 90 degrees, then it will be known as an obtuse angled triangle. Finally, there is an equiangular triangle, in which all the angles of the triangle are 60 degrees. On the other hand, the triangle can also be identified or labeled on the type of sides they have.  A scalene triangle has no congruent sides.  An isosceles triangle has two congruent sides.  An equilateral triangle has three congruent sides. Please note that equilateral and equiangular triangles are the two distinct terms for the same triangle. Polygon Another in the geometric shapes that you need to know about is a polygon. A polygon is made up of only lines and has no curves. It may not have any open parts. In this case, a polygon is basically a broader term to several shapes such as a square, triangle, and a rectangle.

Polygons A polygon is a closed figure where the sides are all line segments. Each side must intersect exactly two others sides but only at their endpoints. The sides must be noncollinear and have a common endpoint. A polygon is usually named after how many sides it has, a polygon with n-sides is called a n-gon. E.g. the building which houses United States Department of Defense is called pentagon since it has 5 sides. Sides Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 9 nonagon 10 decagon A regular polygon is a polygon in which all sides are congruent and all the angles are congruent. Parallelogram A parallelogram is another in the geometric shapes in which the opposite side of the shape are parallel. To be able to examine, if the sides are parallel or not, you’ll have to closely examine the shape. The key property of a parallelogram is that parallel lines

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never cross or intersect each other, no matter how long you extend them. So, if you go on extending the lines through eternity and they never intersect each other, then they can be called a parallelogram.

However, if the lines touch or meet at any given point, then that shape cannot be considered a parallelogram. So, a triangle cannot be considered a parallelogram since the lines opposite to a triangle meet at the point of the triangle. And since the lines intersect, it cannot be called a parallelogram.

Area, perimeter volume formulas L and W are the lengths of the rectangle's sides (length and width). ... Perimeter, Area, and Volume. Table 3. Volume Formulas Shape

Formula

Variables

Cube

V=s3

s is the length of the side.

Right Rectangular Prism

V=LWH

L is the length, W is the width and H is the height.

Prism or Cylinder

V=Ah

A is the area of the base, h is the height.

Formula for geometrical figures pi (π)=3.1415926535 ... Perimeter formula Square

4 × side

Rectangle

2 × (length + width)

Parallelogram

2 × (side1 + side2)

Triangle

side1 + side2 + side3

Regular n-polygon

n × side

Trapezoid

height × (base1 + base2) / 2

Trapezoid

base1 + base2 + height × [csc(theta1) + csc(theta2)]

Circle

2 × pi × radius

Ellipse

4 × radius1 × E(k,pi/2) E(k,pi/2) is the Complete Elliptic Integral of the Second Kind k = (1/radius1) × sqrt(radius12 - radius22)

Area formula Square

side2

Rectangle

length × width

Parallelogram

base × height

Triangle

base × height / 2

Regular n-polygon

(1/4) × n × side2 × cot(pi/n)

Trapezoid

height × (base1 + base2) / 2

Circle

pi × radius2

Ellipse

pi × radius1 × radius2

Cube (surface)

6 × side2

Sphere (surface)

4 × pi × radius2

Cylinder (surface of side) perimeter of circle × height 2 × pi × radius × height Cylinder (whole surface) Areas of top and bottom circles + Area of the side 2(pi × radius2) + 2 × pi × radius × height Cone (surface)

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pi × radius × side

Torus (surface)

pi2 × (radius22 - radius12)

Volume formula Cube

side3

Rectangular Prism

side1 × side2 × side3

Sphere

(4/3) × pi × radius 3

Ellipsoid

(4/3) × pi × radius1 × radius2 × radius3

Cylinder

pi × radius2 × height

Cone

(1/3) × pi × radius2 × height

Pyramid

(1/3) × (base area) × height

Torus

(1/4) × pi2 × (r1 + r2) × (r1 - r2)2

Distance Formula as the Derivative of the Pythagorean Theorem The distance d between two points and is calculated or computed using the following formula:

Below is an illustration showing that the Distance Formula is based on the Pythagorean Theorem where the distance dd is the hypotenuse of a right triangle.

Observations: a) The expression {x_2} - {x_1}x2−x1 is read as the “change in xx“. b) The expression {y_2} - {y_1}y2−y1 is read as the “change in yy“. Examples of Using the Distance Formula Example 1: Find the distance between the two points (–3, 2) and (3, 5). Label the parts of each point properly and substitute into the distance formula. If we let \left( { - 3,2} \right)(−3,2) be the first point then it will take the subscript of 1, thus, {x_1} = - 3x1=−3 and {y_1} = 2y1 =2. Similarly, if \left( {3,5} \right)(3,5) be the second point it will have the subscript of 2, thus, {x_2} = 3x2=3 and {y_2} = 5y2=5. Here is the calculation,

Therefore, the distance between two points (–3, 2) and (3, 5) is 3\sqrt 535 . This is how it looks on a graph.

Example 2: Find the distance between the points (–1, –1) and (4, –5). If we assign \left( { - 1, - 1} \right)(−1,−1) as our first point then

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In the same manner, assigning \left( {4, - 5} \right)(4,−5) as our second point, we have Plugging in the values of xx and yy, we get:

The two points and the distance between them which is \sqrt {41}41 can be shown on a graph just like the one below.

Example 3: Find the distance between the points (–4, –3) and (4, 3). Sometimes you may wonder if switching the points in calculating the distance can affect the final outcome. Well, if you think about it, the formula is squaring the difference of the corresponding xx and yy values. That means it doesn’t matter if the change in xx, also known as delta xx, or the change in yy, also known as delta yy, is negative because when we eventually square it (raise to the 2nd power), the result always comes out to be positive. Let’s “prove” that the answer is always the same by solving this problem two ways! The first solution shows the usual way because we assign which point is the first and second based on the order in which they are given to us in the problem. In the second solution, we switch the points.  Solution 1:



Solution 2:

As you can see, both solutions arrived at the same answer or result which is the distance of 1010, d = 10d=10. Below is the visual solution to the problem.

Example 4: Find the radius of a circle with a diameter whose endpoints are (–7, 1) and (1, 3). Remember that the diameter of a circle is twice the length of its radius. If that’s the case, then the radius is half the length of the diameter.

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Here’s the plan! Since we are given with the endpoints of the diameter, we can use the distance formula to find its length. Finally, we divide it by 2 to get the length of the radius, as required by the problem.  Find the length of the diameter with endpoints \left( { - 7,1} \right)(−7,1) and \left( {1,3} \right)(1,3).

 

Solve for the radius by dividing the diameter by 22. The blue dots are the endpoints of diameter and the green dot is the center of the circle (calculated using the midpoint formula) located at \left( { - 3,2} \right)(−3,2).

Exercises 1. If the distance between the points (5, - 2) and (1, a) is 5, find the values of a. Solution:

We know, the distance between (x1, y1) and (x2, y2)

is (x1−x2)2+(y1−y2)2−−−−−−−−−−−−−−−−−−√ ------Here, the distance = 5, x1 = 5, x2 = 1, y1 = -2 and y2 = a Therefore, 5 = (5−1)2+(−2−a)2−−−−−−−−−−−−−−−−√(5−1)2+(−2−a)2 ⟹ 25 = 16 + (2 + a)22 ⟹ (2 + a)2 = 25 - 16 ⟹ (2 + a)2 = 9 Taking square root, 2 + a = ±3 ⟹ a = -2 ± 3 ⟹ a = 1, -5 2. The co-ordinates of points on the x-axis which are at a distance of 5 units from the point (6, -3). Solution: Let the co-ordinates of the point on the x-axis be (x, 0) Since, distance = (x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√(x2−x1)2+(y2−y1)2 Now taking (6, -3) = (x11, y11) and (x, 0) = (x22, y22), we get 5 = (x−6)2+(0+3)2−−−−−−−−−−−−−−−√(x−6)2+(0+3)2 Squaring both sides we get ⟹ 25 = (x – 6)22 + 322 ⟹ 25 = x22 – 12x + 36 + 9 ⟹ 25 = x22 – 12x + 45 ⟹ x22 – 12x + 45 – 25 = 0 ⟹ x22 – 12x + 20 = 0 ⟹ (x – 2)(x – 10) = 0 ⟹ x = 2 or x = 10 Therefore, the required points on the x-axis are (2, 0) and (10, 0). 3. Which point on the y-axis is equidistance from the points (12, 3) and (-5, 10)? Solution:BBLet the required point on the y-axis (0, y). Given (0, y) is equidistance from (12, 3) and (-5, 10) i.e., distance between (0, y) and (12, 3) = distance between (0, y) and (-5, 10) ⟹ (12−0)2+(3−y)2−−−−−−−−−−−−−−−−√(12−0)2+(3−y)2 = (−5−0)2+(10−y)2−−−−−−−−−−−−−−−−−√(−5−0)2+(10−y)2 ⟹ 144 + 9 + y22 – 6y = 25 + 100 + y22 – 20y ⟹ 14y = -28 ⟹ y = -2 Therefore, the required point on the y-axis = (0, -2) 4. Find the points on the y-axis, each of which is at a distance of 13 units from the point (-5, 7). Solution: Let A (-5, 7) be the given point and let P (0, y) be the required point on the y-axis. Then, PA = 13 units

⟹ PA2 = 169

⟹ (0 + 5)2 + (y - 7)2 = 169

⟹ 25 + y2 - 14y + 49 = 169

⟹ y2 – 14y + 74 = 169

⟹ y2 – 14y – 95 = 0

⟹ (y - 19)(y + 5) = 0

⟹ y – 19 = 0 or, y + 5 = 0

⟹ y = 19 or, y = -5

Hence, the required points are (0, 19) and (0, -5)

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APPLICATIONINSTALLING A Zip line

Supplementary and Complementary Angles Here is a look at two of the relationships. Supplementary angles are two angles that have a sum of 180°.

Complementary angles are two angles that have a sum of 90°.

There is an easy way to try and remember these using the first letters of each word. The S in supplementary can be used to form the 8 in 180.

The C in complementary can be used to form the 9 in 90.

If we know that a set of angles form one of these special relationships, we can determine the measure of the other angle. Example #1: 43° To determine the supplement, subtract the given angle from 180. 180 - 43 = 137° The supplement of 43° is 137°. To determine the complement, subtract the given angle from 90. 90 - 43 = 47° The complement of 43° is 47°. Example #2: 61° 180 - 61 = 119° The supplement of 61° is 119°. 90 - 61 = 29° The complement of 61° is 29°. Example #3: 127° 180 - 127 = 53° The supplement of 127° is 53°. 127° is already greater than 90°. Therefore, there is no complement. Example #4: Determine the missing angle.

Notice that the two angles for a right angle when together. This means that the angles are complementary and have a sum of 90°. 90 - 62 = 28°

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