6'7 Areas of Triangles and Quadrilaterals

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6.7 Areas of Triangles and Quadrilaterals

• Geometry
• Mrs. Spitz
• Spring 2005

2
Objectives

• Find the areas of squares, rectangles,
  parallelograms and triangles.
• Find the areas of trapezoids, kites and rhombuses.

3
Assignment

• pp. 376-378 3-43. Skip 39 and 40.

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Using Area Formulas

• Postulate 22 Area of a Square PostulateThe
  area of a square is the square of the length of
  its side, or A s2.
• Postulate 23 Area Congruence PostulateIf two
  polygons are congruent, then they have the same
  area.
• Postulate 24 Area Addition PostulateThe area
  of a region is the sum of the areas of its
  nonoverlapping parts.

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Area Theorems

• Theorem 6.20Area of a RectangleThe area of a
  rectangle is the product of its base and height.
• You know this one and you have since
  kindergarten. Since you do the others should be
  a breeze.

h
b
A bh
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Area Theorems

• Theorem 6.21Area of a ParallelogramThe area of
  a parallelogram is the product of a base and
  height.

h
b
A bh
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Area Theorems

• Theorem 6.22Area of a TriangleThe area of a
  triangle is one half the product of a base and
  height.

h
b
A ½ bh
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Justification

• You can justify the area formulas for
  parallelograms as follows.
• The area of a parallelogram is the area of a
  rectangle with the same base and height.

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Justification

• You can justify the area formulas for triangles
  follows.
• The area of a triangle is half the area of a
  parallelogram with the same base and height.

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Ex. 1 Using the Area Theorems

• Find the area of ?ABCD.
• Solution
• Method 1 Use AB as the base. So, b16 and h9
• Areabh16(9) 144 square units.
• Method 2 Use AD as the base. So, b12 and h12
• Areabh12(12) 144 square units.
• Notice that you get the same area with either
  base.

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Ex. 2 Finding the height of a Triangle

• Rewrite the formula for the area of a triangle in
  terms of h. Then use your formula to find the
  height of a triangle that has an area of 12 and a
  base length of 6.
• Solution
• Rewrite the area formula so h is alone on one
  side of the equation.
• A ½ bh Formula for the area of a triangle
• 2Abh Multiply both sides by 2.
• 2Ah Divide both sides by b.
• b
• Substitute 12 for A and 6 for b to find the
  height of the triangle.
• h2A 2(12) 24 4
• b 6 6
• ?The height of the triangle is 4.

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Ex. 3 Finding the Height of a Triangle

• A triangle has an area of 52 square feet and a
  base of 13 feet. Are all triangles with these
  dimensions congruent?
• Solution Using the formula from Ex. 2, the
  height is
• h 2(52) 104 8
• 13 13
• Here are a few triangles with these dimensions

8
8
8
8
13
13
13
13
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Areas of Trapezoids

• Theorem 6.23Area of a TrapezoidThe area of a
  trapezoid is one half the product of the height
  and the sum of the bases.
• A ½ h(b1 b2)

b1
h
b2
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Areas of Kites
d1

• Theorem 6.24Area of a KiteThe area of a kite is
  one half the product of the lengths of its
  diagonals.
• A ½ d1d2

d2
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Areas of Rhombuses

• Theorem 6.24Area of a RhombusThe area of a
  rhombus is one half the product of the lengths of
  the diagonals.
• A ½ d1 d2

d2
d1
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Areas of Trapezoids, Kites and Rhombuses

• You will have to justify theorem 6.23 in
  Exercises 58 and 59. You may find it easier to
  remember the theorem this way.

b1
h
b2
Height
Area
Length of Midsegment
x

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Ex. 4 Finding the Area of a Trapezoid

• Find the area of trapezoid WXYZ.
• Solution The height of WXYZ is h5 1 4
• Find the lengths of the bases.
• b1 YZ 5 2 3
• b2 XW 8 1 7

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Ex. 4 Finding the Area of a Trapezoid
Substitute 4 for h, 3 for b1, and 7 for b2 to
find the area of the trapezoid. A ½ h(b1 b2)
Formula for area of a trapezoid. A ½ (4)(3 7
) Substitute A ½ (40) Simplify A 20 Simplify
?The area of trapezoid WXYZ is 20 square units
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Justification of Kite/Rhombuses formulas

• The diagram at the right justifies the formulas
  for the areas of kites and rhombuses. The
  diagram show that the area of a kite is half the
  area of a rectangle whose length and width are
  the lengths of the diagonals of the kite. The
  same is true for a rhombus.

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Ex. 5 Finding the area of a rhombus
B

• Use the information given in the diagram to find
  the area of rhombus ABCD.
• Solution
• Method 1 Use the formula for the area of a
  rhombus d1 BD 30 and d2 AC 40

15
20
20
A
C
24
15
D
E
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Ex. 5 Finding the area of a rhombus
B

• A ½ d1 d2
• A ½ (30)(40)
• A ½ (120)
• A 60 square units
• Method 2 Use the formula for the area of a
  parallelogram, b25 and h 24.
• A bh 25(24) 600 square units

15
20
20
A
C
24
15
D
E
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• ROOF Find the area of the roof. G, H, and K are
  trapezoids and J is a triangle. The hidden back
  and left sides of the roof are the same as the
  front and right sides.

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SOLUTION

• Area of J ½ (20)(9) 90 ft2.
• Area of G ½ (15)(2030) 375 ft2.
• Area of J ½ (15)(4250) 690 ft2.
• Area of J ½ (12)(3042) 432 ft2.
• The roof has two congruent faces of each type.
• Total area2(90375690432)3174
• ?The total area of the roof is 3174 square feet.