CONFIDENTIAL 1 Geometry Developing Formulas for Triangles and Quadrilaterals
CONFIDENTIAL1 Geometry Developing Formulas for Triangles and Quadrilaterals.
Dec 21, 2015
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- Slide 1
- CONFIDENTIAL1 Geometry Developing Formulas for Triangles and Quadrilaterals
- Slide 2
- CONFIDENTIAL2 Warm up Find the perimeter and area of each figure: 1) 2) 1)P = 6x + 4; A = 2x 2 + 4x 2) P = 2x + 1; A = 7x/2 x + 2 2x x x + 1 7
- Slide 3
- CONFIDENTIAL3 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.
- Slide 4
- CONFIDENTIAL4 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
- Slide 5
- CONFIDENTIAL5 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 = s 2, and perimeter is P = 4s.
- Slide 6
- CONFIDENTIAL6 Finding measurements of Parallelograms Find each measurement: A) the area of a Parallelogram 6 in h 5 in 3 in Step 1: Use Pythagorean Theorem to find the height h. 3 2 + h 2 = 5 2 h = 4 Step 2: Use h to find the area of parallelogram. A = bh A = 6(4) A = 24 in 2 Area of a parallelogram. Substitute 6 for b and 4 for h. Simplify.
- Slide 7
- CONFIDENTIAL7 B) the height of a rectangle in which b = 5 cm and A = (5x 2 5x) cm 2. A = bh 5x 2 5x = 5(h) 5(x 2 x) = 5(h) x 2 x = h h = (x 2 x) cm Area of a rectangle. Substitute (5x 2 5x) for A and 5 for b. Factor 5 out of the expression for A. Divide both sides by 5. Sym. Prop. of =.
- Slide 8
- CONFIDENTIAL8 C) the perimeter of a rectangle in which A = 12x ft 2. 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
- Slide 9
- CONFIDENTIAL9 Now you try! 1) Find the base of a Parallelogram in which h = 65 yd and A = 28 yd 2. 1) b = 0.5 yd
- Slide 10
- CONFIDENTIAL10 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 bb b1b1 b2b2 h b2b2 b2b2 h b1b1
- Slide 11
- CONFIDENTIAL11 Area: Triangles and Trapezoids The area of a Triangle with base b and height h is A = 1 bh. 2 h b The area of a Trapezoid with bases b 1 and b 2 and height h is A = 1 (b 1 + b 2 )h. b2b2 h b1b1 2
- Slide 12
- CONFIDENTIAL12 Finding measurements if Triangles and Trapezoids Find each measurement: A) the area of Trapezoid with b 1 = 9 cm, b 2 = 12 cm and h = 3 cm. A = 1 (b 1 + b 2 )h 2 Area of a Trapezoid. Substitute 9 for b 1, 12 for b 2 and 3 for h. Simplify. A = 1 (9 + 12 )3 2 A = 31.5 cm 2
- Slide 13
- CONFIDENTIAL13 B) the base of Triangle in which A = x 2 in 2. A = 1 bh 2 Area of a Triangle. Substitute x 2 for A and x for h. Divide both sides by x. x 2 = 1 bx 2 x = 1 b 2 x in b b = 2x in. 2x = b Multiply both sides by 2. Sym. Prop. of =.
- Slide 14
- CONFIDENTIAL14 C) b 2 of the Trapezoid in which A = 8 ft 2. Multiply by 2. 8 = 3 + b 2 b 2 = 5 ft. 5 = b 2 Subtract 3 from both sides. Sym. Prop. of =. A = 1 (b 1 + b 2 )h 2 Area of a Trapezoid. Substitute 8 for A 1, 3 for b 1 and 2 for h. 8 = 1 (3 + b 2 )2 2 3 ft 2 ft b2b2
- Slide 15
- CONFIDENTIAL15 Now you try! 2) Find the area of the triangle. 2) b = 96 m 2 20 m 12 m b
- Slide 16
- CONFIDENTIAL16 A kite or a rhombus with diagonal d 1 and d 2 can be divided into two congruent triangles with a base d 1 and height of d 2. d1d1 d 2 d1d1 Total area : A = 2(1 d 1 d 2 ) = 1 d 1 d 2 4 2 area of each triangle: A = 1 d 1 ( d 2 ) 2 = 1 d 1 d 2 4
- Slide 17
- CONFIDENTIAL17 Area: Rhombus and kites The area of a rhombus or kite with diagonals d 1 and d 2 and height h is A = 1 d 1 d 2. 2 d1d1 d 2 d1d1
- Slide 18
- CONFIDENTIAL18 Finding measurements of Rhombus and kites Find each measurement: A) d 2 of a kite with d 1 = 16 cm, and A = 48 cm 2. A = 1 (d 1 d 2 ) 2 Area of a kite. Substitute 48 for A, 16 for d 1. Simplify. 48 = 1 (16)d 2 2 d 2 = 6 cm
- Slide 19
- CONFIDENTIAL19 B) The area of the rhombus. A = 1 (d 1 d 2 ) 2 Area of a kite. Substitute (6x + 4) for d 1 and (10x + 10) for d 2. Multiply the binomials. A = 1 (6x + 4) (10x + 10) 2 d 1 = (6x + 4)in. d 2 = (10x + 10)in. A = 1 (6x 2 + 100x + 40) 2 Simplify. A = (3x 2 + 50x + 20)
- Slide 20
- 41 ft 9 ft 15 ft y x CONFIDENTIAL20 C) The area of the kite. Step 1: The diagonal d 1 and d 2 form four right angles. Use Pythagorean Theorem to find the x and y. 9 2 + x 2 = 41 2 x 2 = 1600 x = 40 9 2 + y 2 = 15 2 y 2 = 144 y = 12
- Slide 21
- CONFIDENTIAL21 41 ft 9 ft 15 ft y x Step 2: Use d 1 and d 2 to find the area. d 1 = (x + y) which is 52. Half of d 2 = 9, so d 2 = 18. A = 1 (d 1 d 2 ) 2 Area of a kite. Substitute 52 for d 1 and 18 for d 2. A = 1 (52) (18) 2 Simplify. A = 468 ft 2
- Slide 22
- CONFIDENTIAL22 Now you try! 3) b = 96 m 2 3) Find d 2 of a rhombus with d 1 = 3x m, and A = 12xy m 2.
- Slide 23
- CONFIDENTIAL23 Games Application The 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.
- Slide 24
- CONFIDENTIAL24 A = 1 (d 1 d 2 ) = 1 (2)(2) = 2 cm 2. 2 Area: Method 1: d 2 of a kite with d 1 = 16 cm, and A = 48 cm 2. Method 2: The side length of the red square is 2 cm, so the area if A = (s 2 ) = (2) 2 = 2 cm 2.
- Slide 25
- CONFIDENTIAL25 Now you try! 4) A = 4 cm 2 P = 4 + 42 cm 4) Find the area and perimeter of the large yellow triangle in the figure given below.
- Slide 26
- CONFIDENTIAL26 Now some problems for you to practice !
- Slide 27
- CONFIDENTIAL27 Find each measurement: Assessment 1)120 cm 2 2) 5x ft 12 cm 10 cm 1) the area of the Parallelogram. 2x ft 2) the height of the rectangle in which A = 10x 2 ft 2.
- Slide 28
- CONFIDENTIAL28 Find each measurement: 3) 240 m 2 4) 13 in 3) the area of the Trapezoid. 4) the base of the triangle in which A = 58.5 in 2. 20 m 9 m 15 m 9 in
- Slide 29
- CONFIDENTIAL29 5) 175 in 2 6) 25 m 5) the area of the rhombus. 6) d 2 of the kite in which A = 187.5 m 2. Find each measurement: 14 in 25 in 15 m
- Slide 30
- CONFIDENTIAL30 7) The rectangle with perimeter of (26x + 16) cm and an area of (42x 2 + 51x + 15) cm 2. Find the dimensions of the rectangle in terms of x. 7) (7x + 5) and (6x + 3)
- Slide 31
- CONFIDENTIAL31 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 ft 2
- Slide 32
- CONFIDENTIAL32 Lets 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.
- Slide 33
- CONFIDENTIAL33 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 = s 2, and perimeter is P = 4s.
- Slide 34
- CONFIDENTIAL34 Finding measurements of Parallelograms Find each measurement: A) the area of a Parallelogram 6 in h 5 in 3 in Step 1: Use Pythagorean Theorem to find the height h. 3 2 + h 2 = 5 2 h = 4 Step 2: Use h to find the area of parallelogram. A = bh A = 6(4) A = 24 in 2 Area of a parallelogram. Substitute 6 for b and 4 for h. Simplify.
- Slide 35
- CONFIDENTIAL35 Area: Triangles and Trapezoids The area of a Triangle with base b and height h is A = 1 bh. 2 h b The area of a Trapezoid with bases b 1 and b 2 and height h is A = 1 (b 1 + b 2 )h. b2b2 h b1b1 2
- Slide 36
- CONFIDENTIAL36 Finding measurements if Triangles and Trapezoids Find each measurement: A) the area of Trapezoid with b 1 = 9 cm, b 2 = 12 cm and h = 3 cm. A = 1 (b 1 + b 2 )h 2 Area of a Trapezoid. Substitute 9 for b 1, 12 for b 2 and 3 for h. Simplify. A = 1 (9 + 12 )3 2 A = 31.5 cm 2
- Slide 37
- CONFIDENTIAL37 A kite or a rhombus with diagonal d 1 and d 2 can be divided into two congruent triangles with a base d 1 and height of d 2. d1d1 d 2 d1d1 Total area : A = 2(1 d 1 d 2 ) = 1 d 1 d 2 4 2 area of each triangle: A = 1 d 1 ( d 2 ) 2 = 1 d 1 d 2 4
- Slide 38
- CONFIDENTIAL38 Area: Rhombus and kites The area of a rhombus or kite with diagonals d 1 and d 2 and height h is A = 1 d 1 d 2. 2 d1d1 d 2 d1d1
- Slide 39
- CONFIDENTIAL39 Finding measurements of Rhombus and kites Find each measurement: A) d 2 of a kite with d 1 = 16 cm, and A = 48 cm 2. A = 1 (d 1 d 2 ) 2 Area of a kite. Substitute 48 for A, 16 for d 1. Simplify. 48 = 1 (16)d 2 2 d 2 = 6 cm
- Slide 40
- CONFIDENTIAL40 Games Application The 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.
- Slide 41
- CONFIDENTIAL41 A = 1 (d 1 d 2 ) = 1 (2)(2) = 2 cm 2. 2 Area: Method 1: d 2 of a kite with d 1 = 16 cm, and A = 48 cm 2. Method 2: The side length of the red square is 2 cm, so the area if A = (s 2 ) = (2) 2 = 2 cm 2.
- Slide 42
- CONFIDENTIAL42 You did a great job today!
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