CONFIDENTIAL 1 Geometry Developing Formulas for Triangles and Quadrilaterals
Geometry Developing Formulas for Triangles and Quadrilaterals
Feb 07, 2016
Geometry Developing Formulas for Triangles and Quadrilaterals. Warm up. Find the perimeter and area of each figure: 1) 2). x + 1. 2x. 7. x. x + 2. P = 6x + 4; A = 2x 2 + 4x 2) P = 2x + 1; A = 7x/2. Area Addition Postulate. - PowerPoint PPT Presentation
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CONFIDENTIAL 1
Geometry
Developing Formulas for Triangles and
Quadrilaterals
CONFIDENTIAL 2
Warm up
Find the perimeter and area of each figure:
1) 2)
1) P = 6x + 4; A = 2x2 + 4x 2) P = 2x + 1; A = 7x/2
x + 2
2x
x
x + 17
CONFIDENTIAL 3
When a Figure is made from different shapes, the area of the figure is the sum of the areas of the pieces.
Area Addition Postulate
Postulate 1: The area of a region is equal to the sum of the areas of non-overlapping parts.
CONFIDENTIAL 4
Recall that a rectangle with base b and height h has an area of A = bh. You can use the Area Addition Postulate
to see that a parallelogram has the same area as a rectangle with the same base and height.
b
h
b
CONFIDENTIAL 5
The area of a Parallelogram with base b and height h is A = bh.
Area: Parallelogram
b
h
Remember that rectangles and squares are also Parallelograms. The area of a square with side s is
A = s2, and perimeter is P = 4s.
CONFIDENTIAL 6
Finding measurements of Parallelograms Find each measurement:
A) the area of a Parallelogram
6 in
h5 in
3 inStep 1: Use Pythagorean Theorem to find the height h.
32 + h2 = 52
h = 4
Step 2: Use h to find the area of parallelogram.
A = bhA = 6(4)A = 24 in2
Area of a parallelogram.Substitute 6 for b and 4 for h.Simplify.
CONFIDENTIAL 7
B) the height of a rectangle in which b = 5 cm and A = (5x2 – 5x) cm2.
A = bh
5x2 – 5x = 5(h)
5(x2 – x) = 5(h)
x2 – x = h
h = (x2 – x) cm
Area of a rectangle .
Substitute (5x2 – 5x) for A and 5 for b.
Factor 5 out of the expression for A.
Divide both sides by 5.
Sym. Prop. of =.
CONFIDENTIAL 8
C) the perimeter of a rectangle in which A = 12x ft2.
A = bh
12x = 6(b)
2x = b
Area of a rectangle .
Substitute 12x for A and 6 for h.
Divide both sides by 6.
Perimeter of a rectangle
Substitute 2x for A and 6 for h.
Step 1: Use Pythagorean Theorem to find the height h.
P= 2b + 2h
P = 2(2x) +2(6)
P = (4x +12) ft
Step 2: Use the base and height to find the perimeter.
Simplify.
6x
CONFIDENTIAL 9
Now you try!
1) Find the base of a Parallelogram in which h = 65 yd and A = 28 yd2.
1) b = 0.5 yd
CONFIDENTIAL 10
To understand the formula for the area of a triangle or trapezoid, notice that the two congruent triangles or
two congruent trapezoids fit together to form a parallelogram. Thus the area of a triangle or a
trapezoid is half the area of the related parallelogram.
h h
b b
b1
b2
h
b2
b2
h
b1
CONFIDENTIAL 11
Area: Triangles and Trapezoids
The area of a Triangle with base b and height h is A = 1 bh.
2h
b
The area of a Trapezoid with bases b1 and b2 and height h is
A = 1 (b1 + b2 )h.b2
h
b1
2
CONFIDENTIAL 12
Finding measurements if Triangles and Trapezoids
Find each measurement:
A) the area of Trapezoid with b1 = 9 cm, b2 = 12 cm and h = 3 cm.
A = 1 (b1 + b2 )h 2
Area of a Trapezoid.
Substitute 9 for b1, 12 for b2 and 3 for h.
Simplify.
A = 1 (9 + 12 )3 2
A = 31.5 cm2
CONFIDENTIAL 13
B) the base of Triangle in which A = x2 in2.
A = 1 bh 2
Area of a Triangle.
Substitute x2 for A and x for h.
Divide both sides by x.
x2 = 1 bx 2
x = 1 b 2
x in
b
b = 2x in.
2x = b Multiply both sides by 2.
Sym. Prop. of =.
CONFIDENTIAL 14
C) b2 of the Trapezoid in which A = 8 ft2.
Multiply ½ by 2.8 = 3 + b2
b2 = 5 ft.
5 = b2 Subtract 3 from both sides.
Sym. Prop. of =.
A = 1 (b1 + b2 )h 2
Area of a Trapezoid.
Substitute 8 for A1, 3 for b1 and 2 for h.8 = 1 (3 + b2)2 2
3 ft
2 ft
b2
CONFIDENTIAL 15
Now you try!
2) Find the area of the triangle.
2) b = 96 m2
20 m12 m
b
CONFIDENTIAL 16
A kite or a rhombus with diagonal d1 and d2 can be divided into two congruent triangles with a base d1
and height of ½ d2 .
d1
½ d2
d1
½ d2
Total area : A = 2(1 d1d2 ) = 1 d1d2
4 2
area of each triangle: A = 1 d1(½ d2 ) 2 = 1 d1d2
4
CONFIDENTIAL 17
Area: Rhombus and kites
The area of a rhombus or kite with diagonals d1 and d2 and height h is A = 1 d1d2 .
2
d1
½ d2 d1
½ d2
CONFIDENTIAL 18
Finding measurements of Rhombus and kites
Find each measurement:
A) d2 of a kite with d1 = 16 cm, and A = 48 cm2.
A = 1 (d1d2 ) 2
Area of a kite.
Substitute 48 for A, 16 for d1.
Simplify.
48 = 1 (16)d2
2
d2 = 6 cm
CONFIDENTIAL 19
B) The area of the rhombus .
A = 1 (d1d2 ) 2
Area of a kite.
Substitute (6x + 4) for d1 and (10x + 10) for d2.
Multiply the binomials.
A = 1 (6x + 4) (10x + 10) 2
d1 = (6x + 4)in.
d2 = (10x + 10)in.
A = 1 (6x2 + 100x + 40) 2
Simplify.A = (3x2 + 50x + 20)
41 ft9 ft
15 ftyx
CONFIDENTIAL 20
C) The area of the kite.
Step 1: The diagonal d1 and d2 form four right angles.
Use Pythagorean Theorem to find the x and y.
92 + x2 = 412
x2 = 1600
x = 40
92 + y2 = 152
y2 = 144
y = 12
CONFIDENTIAL 21
41 ft9 ft
15 ftyx
Step 2: Use d1 and d2 to find the area. d1 = (x + y) which is 52. Half of d2 = 9, so d2 = 18.
A = 1 (d1d2 ) 2
Area of a kite.
Substitute 52 for d1 and 18 for d2.A = 1 (52) (18) 2
Simplify.A = 468 ft2
CONFIDENTIAL 22
Now you try!
3) b = 96 m2
3) Find d2 of a rhombus with d1 = 3x m, and A = 12xy m2.
CONFIDENTIAL 23
Games ApplicationThe pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the
red square.
Perimeter:
Each side of the red square is the diagonal of the square grid. Each grid
square has a side length of 1 cm, so the diagonal is √2 cm. The perimeter of the
red square is P = 4s = 4 √2 cm.
CONFIDENTIAL 24
A = 1 (d1d2 ) = 1 (√2)(√2) = 2 cm2. 2 2
Area:
Method 1: d2 of a kite with d1 = 16 cm, and A = 48 cm2.
Method 2: The side length of the red square is √2 cm, so the area if A = (s2) = (√2)2 = 2 cm2.
CONFIDENTIAL 25
Now you try!
4) A = 4 cm2 P = 4 + 4√2 cm
4) Find the area and perimeter of the large yellow triangle in the figure given below.
CONFIDENTIAL 26
Now some problems for you to practice !
CONFIDENTIAL 27
Find each measurement:
Assessment
1) 120 cm2
2) 5x ft
12 cm
10 cm1) the area of the Parallelogram.
2x ft
2) the height of the rectangle in which A = 10x2 ft2.
CONFIDENTIAL 28
Find each measurement:
3) 240 m2
4) 13 in
3) the area of the Trapezoid.
4) the base of the triangle in which A = 58.5 in2.
20 m
9 m 15 m
9 in
CONFIDENTIAL 29
5) 175 in2
6) 25 m
5) the area of the rhombus.
6) d2 of the kite in which A = 187.5 m2.
Find each measurement:
14 in
25 in
15 m
CONFIDENTIAL 30
7) The rectangle with perimeter of (26x + 16) cm and an area of (42x2 + 51x + 15) cm2. Find the dimensions
of the rectangle in terms of x.
7) (7x + 5) and (6x + 3)
CONFIDENTIAL 31
8) The stained-glass window shown ii a rectangle with a base of 4 ft and a height of 3 ft. Use the grid to find
the area of each piece.
8) √10 ft2
CONFIDENTIAL 32
Let’s review
When a Figure is made from different shapes, the area of the figure is the sum of the areas of the pieces.
Postulate 1: The area of a region is equal to the sum of the areas of non-overlapping parts.
CONFIDENTIAL 33
The area of a Parallelogram with base b and height h is A = bh.
Area: Parallelogram
b
h
Remember that rectangles and squares are also Parallelograms. The area of a square with side s is
A = s2, and perimeter is P = 4s.
CONFIDENTIAL 34
Finding measurements of Parallelograms Find each measurement:
A) the area of a Parallelogram
6 in
h5 in
3 inStep 1: Use Pythagorean Theorem to find the height h.
32 + h2 = 52
h = 4
Step 2: Use h to find the area of parallelogram.
A = bhA = 6(4)A = 24 in2
Area of a parallelogram.Substitute 6 for b and 4 for h.Simplify.
CONFIDENTIAL 35
Area: Triangles and Trapezoids
The area of a Triangle with base b and height h is A = 1 bh.
2h
b
The area of a Trapezoid with bases b1 and b2 and height h is
A = 1 (b1 + b2 )h.b2
h
b1
2
CONFIDENTIAL 36
Finding measurements if Triangles and Trapezoids
Find each measurement:
A) the area of Trapezoid with b1 = 9 cm, b2 = 12 cm and h = 3 cm.
A = 1 (b1 + b2 )h 2
Area of a Trapezoid.
Substitute 9 for b1, 12 for b2 and 3 for h.
Simplify.
A = 1 (9 + 12 )3 2
A = 31.5 cm2
CONFIDENTIAL 37
A kite or a rhombus with diagonal d1 and d2 can be divided into two congruent triangles with a base d1
and height of ½ d2 .
d1
½ d2
d1
½ d2
Total area : A = 2(1 d1d2 ) = 1 d1d2
4 2
area of each triangle: A = 1 d1(½ d2 ) 2 = 1 d1d2
4
CONFIDENTIAL 38
Area: Rhombus and kites
The area of a rhombus or kite with diagonals d1 and d2 and height h is A = 1 d1d2 .
2
d1
½ d2 d1
½ d2
CONFIDENTIAL 39
Finding measurements of Rhombus and kites
Find each measurement:
A) d2 of a kite with d1 = 16 cm, and A = 48 cm2.
A = 1 (d1d2 ) 2
Area of a kite.
Substitute 48 for A, 16 for d1.
Simplify.
48 = 1 (16)d2
2
d2 = 6 cm
CONFIDENTIAL 40
Games ApplicationThe pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the
red square.
Perimeter:
Each side of the red square is the diagonal of the square grid. Each grid
square has a side length of 1 cm, so the diagonal is √2 cm. The perimeter of the
red square is P = 4s = 4 √2 cm.
CONFIDENTIAL 41
A = 1 (d1d2 ) = 1 (√2)(√2) = 2 cm2. 2 2
Area:
Method 1: d2 of a kite with d1 = 16 cm, and A = 48 cm2.
Method 2: The side length of the red square is √2 cm, so the area if A = (s2) = (√2)2 = 2 cm2.
CONFIDENTIAL 42
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