highschool/Grade 9/Math/MPM1DZ/Unit 4: Measurement and Geometry.md

Unit 4: Measurement and Geometry

Angle Theorems

1. Transversal Parallel Line Theorems (TPT)
   1. Alternate Angles are Equal (Z-Pattern)
      
   2. Corresponding Angles Equal (F-Pattern)
      
   3. Interior Angles add up to 180 (C-Pattern)

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2. Supplementary Angle Triangle (SAT)
   

• When two angles add up to 180 degrees

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3. Opposite Angle Theorem (OAT) (OAT)
   

• Two lines intersect, two angles form opposite. They have equal measures

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4. Complementary Angle Theorem (CAT)
   

• The sum of two angles that add up to 90 degrees

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5. Angle Sum of a Triangle Theorem (ASTT)
   

• The sum of the three interior angles of any triangle is 180 degrees

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6. Exterior Angle Theorem (EAT)

• The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles

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7. Isosceles Triangle Theorem (ITT)
   

• The base angles in any isosceles triangle are equal

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8. Sum of The Interior Angle of a Polygon

• The sum of the interioir angles of any polygon is 180(n-2) or 180n - 360, where n is the number of sides of the polygon

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9. Exterior Angles of a Convex Polygon

• The sum of the exterior angle of any convex polygon is always 360 degrees

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Properties of Quadrilaterals

• Determine the shape using the properties of it

Figure	Properties
Scalene Triangle	no sides equal
Isosceles Triangle	two sides equal
Equilateral Triangle	All sides equal
Right Angle Triangle	Two sides are perpendicular to each other
Parallelogram	Opposite sides are parallel and have equal length. Additionally, the diagonals bisect each other
Rectangle	Adjacent sides are perpendicular to each other. Furthermore, the diagonals bisect each other and are equal in length
Square	All sides are equal in length. The adjacent sides and diagonals are perpendicular. The adjacent sides are equal in length, so as the diagonals
Rhombus	Opposite sides are parallel and all sides are equal to each other, the diagonals are perpendicular
Trapezoid	There is one pair of opposite sides and they are parallel and unequal in length
Kite	The diagonals are perpendicular

2D Geometry Equations

Shape	Formula	Picture
Rectangle	Area: \(`lw`\) Perimeter: \(`2(l+w)`\)
Triangle	Area: \(`\frac{bh}{2}`\) Perimeter: \(`a+b+c`\)
Circle	Area: \(`πr^2`\) Circumference: \(`2πr`\) or \(`πd`\)
Trapezoid	Area: \(` \frac{(a+b)h}{2}`\) Perimeter: \(`a+b+c+d`\)

3D Geometry Equations

3D Object	Formula	Picture
Rectangular Prism	Volume: \(`lwh`\) SA: \(`2(lw+lh+wh)`\)
Square Based Pyramid	Volume: \(`\frac{1}{3} b^2 h`\) SA: \(`2bs+b^2`\)
Sphere	Volume: \(`\frac{4}{3} πr^3`\) SA: \(`4πr^2`\)
Cone	Volume: \(` \frac{1}{3} πr^2 h`\) SA: \(`πrs+πr^2`\)
Cylinder	Volume: \(`πr^2h`\) SA: \(`2πr^2+2πh`\)
Triangular Prism	Volume: \(`ah+bh+ch+bl`\) SA: \(` \frac{1}{2} blh`\)

Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area)

2D Objects

Shape	Maximum Area	Minimum Perimeter
4-sided rectangle	A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width \(`A = lw`\) \(`A_{max} = (w)(w)`\) \(`A_{max} = w^2`\)	A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width. \(`P = 2(l+w)`\) \(`P_{min} = 2(w + w)`\) \(`P_{min} = 2(2w)`\) \(`P_{min} = 4w`\)
3-sided rectangle	\(`l = 2w`\) \(`A = lw`\) \(`A_{max} = 2w(w)`\) \(`A_{max} = 2w^2`\)	\(`l = 2w`\) \(`P = l+2w`\) \(`P_{min} = 2w+2w`\) \(`P_{min} = 4w`\)

3D Objects

3D Object	Maximum Volumne	Minimum Surface Area
Cylinder(closed-top)	The cylinder must be similar to a cube where \(`h = 2r`\) \(`V = πr^2h`\) \(`V_{max} = πr^2(2r)`\) \(`V_{max} = 2πr^3`\)	The cylinder must be similar to a cube where \(`h = 2r`\) \(`SA = 2πr^2+2πrh`\) \(`SA_{min} = 2πr^2+2πr(2r)`\) \(`SA_{min} = 2πr^2+4πr^2`\) \(`SA_{min} = 6πr^2`\)
Rectangular Prism(closed-top)	The prism must be a cube, where \(`l = w = h`\) \(`V = lwh`\) \(`V_{max} = (w)(w)(w)`\) \(`V_{max} = w^3`\)	The prism must be a cube, where \(`l = w = h`\) \(`SA = 2lh+2lw+2wh`\) \(`SA_{min} = 2w^2+2w^2+2w^2`\) \(`SA_{min} = 6w^2`\)
Cylinder(open-top)	\(`h = r`\) \(`V = πr^2h`\) \(`V_{max} = πr^2(r)`\) \(`V_{max} = πr^3`\)	\(`h = r`\) \(`SA = πr^2+2πrh`\) \(`SA_{min} = πr^2+2πr(r)`\) \(`SA_{min} = πr^2+2πr^2`\) \(`SA_{min} = 3πr^2`\)
Square-Based Rectangular Prism(open-top)	\(`h = \frac{w}{2}`\) \(`V = lwh`\) \(`V_{max} = (w)(w)(\frac{w}{2})`\) \(`V_{max} = \frac{w^3}{2}`\)	\(`h = \frac{w}{2}`\) \(`SA = w^2+4wh`\) \(`SA_{min} = w^2+4w(\frac{w}{2})`\) \(`SA_{min} = w^2+2w^2`\) \(`SA_{min} = 3w^2`\)

Labelling

• Given any polygons, labelling the vertices must always:

  1. use CAPITAL LETTERS
  2. they have to be labeled in clockwise or counter-clockwise directions
     

• For a triangle, the side lengths are labeled in LOWERCASE LETTERS associated to the opposite side of the vertex

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Median

• Each median divides the triangle into 2 smaller triangles of equal area
  

• The centroid is exactly \(`\dfrac{2}{3}`\)the way of each median from the vertex, or \(`\dfrac{1}{3}`\) the way from the midpoint of the opposite side, or 2:1 ratio
  

• The three medians divide the triangle into 6 smaller triangles of equal area and 3 pairs of congruent triangles

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Terms:

• Altitude The height of a triangle, a line segment through a vertex and perpendicular to the opposite side
  
• Orthocenter: where all 3 altitudes of the triangle intersect
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• Midpoint: A point on a line where the length of either side of the point are equal
  
• Median: A line segment joining the vertex to the midpoint of the opposite side
  
• Midsegment: A line joining 2 midpoints of the 2 sides of a triangle
  
• Centroid: The intersection of the 3 medians of a triangle
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Proportionality theorem:

• The midsegment of a triangle is half the length of the opposite side and parallel to the opposite side
  
• Three midsegment of a triangle divide 4 congruent triangles with the same area
  
• The Ratio of the outer triangle to the triangle created by the 3 midsegments is 4 to 1
  
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Tips

• Make sure to know your optimization formualas
  
• Read the word problems carefully, determine which formual to use
  
• Never ASSUME, be sure to CALCULATE as most of the time the drawings are NOT ACCURATE
  
• To find missing area, take what you have, subtract what you don’t want
  
• Don’t be afraid to draw lines to help you solve the problem