Area Topic 13 Lesson 5 Effects of Changing Dimensions Proportionally Holt Geometry Texas ©2007

Objectives and Student Expectations TEKS Focus: (10)(B) Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional changes. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (11)(A) Apply the formula for the area of regular polygons to solve problems using appropriate units of measure.

Find the perimeter and area of the rectangle. Example: 1 Find the perimeter and area of the rectangle. Length Width Perimeter P = 2(4) +2(10) P = 28 cm Area A = 10(4) A = 40 cm2

Find the perimeter and area of the rectangle if the length is tripled. Example: 2 Find the perimeter and area of the rectangle if the length is tripled. Perimeter P = 2(4) +2(30) P = 68 cm Area A = 30(4) A = 120 cm2

Example: 3 Perimeter P = 2(16) +2(40) P = 112 cm Area A = 40(16) Find the perimeter and area of the rectangle if the length and width is multiplied by 4. Perimeter P = 2(16) +2(40) P = 112 cm Area A = 40(16) A = 640 cm2

Data from Examples 1-3. Rectangle Original Dimensions Only Length Tripled Length & Width Quadrupled Perimeter 28 cm 68 cm 112 cm Area 40 cm2 120 cm2 640 cm2 How does the area change if only the length is tripled? Area is multiplied by 3 If both length and width are quadrupled, how does the perimeter change? How does the area change? Perimeter is multiplied by 4 Area is multiplied by 16 or 42

Find the perimeter and area of the triangle. Example: 4 Find the perimeter and area of the triangle. Perimeter P = 15+14+13 P = 42 in Area A = A = 84 in2

Example: 5 Find the perimeter and area of the triangle if all dimensions are multiplied by 5. Perimeter P = 75+70+65 P = 210 in Area A = A = 2100 in2

Data from Examples 4-5. Triangle Original Dimensions All Dimensions Multiplied by 5 Perimeter 42 in 210 in Area 84 in2 2100 in2 If all dimensions are multiplied by 5, is the height also multiplied by 5? Yes If all dimensions are multiplied by 5, how does the perimeter change? How does the area change? Perimeter is multiplied by 5 Area is multiplied by 25 or 52

Find the circumference and area of the circle. Example: 6 Find the circumference and area of the circle. Circumference C = 2(8) C = 16 yd Area A = (8) 2 A = 64  yd2

Example: 7 Find the circumference and area of the circle, if the radius is multiplied by 6. 6 Circumference C = 2(48) C = 96 yd Area A = (48) 2 A = 2304 yd2

Data from Examples 6-7. Circle Original Dimensions Radius multiplied by 6 Perimeter 16 yd 96 yd Area 64 yd2 2304 yd2 If the radius is multiplied by 6, how does the circumference change? Circumference multiplied by 6 If the radius is multiplied by 6, how does the area change? Area is multiplied by 36 or 62

Conjectures: If only one dimension of a two dimensional figure is changed, then the two figures (original and new figure) are NOT _________. If all dimensions of a two dimensional figure are changed, then the two figures (original and new figure) are __________. If all dimensions of a two dimensional figure are changed by a factor of a, then the perimeter changes by a factor of ____ and the area changes by a factor of ____. similar similar a a2

Describe the effect of each change on the area of the given figure. Example: 8 & 9 Describe the effect of each change on the area of the given figure. 8. If the height of a triangle is doubled, then the area is ________. 9. If only the height of a trapezoid with base lengths 12 cm and 18 cm and height 5 cm is multiplied by 1/3, then the area is multiplied by ________________. doubled 1/3

Example: 10 & 11 Describe the effect of each change on the perimeter or circumference and the area of the given figure. 10. If the base and height of a triangle with base 12 in and height 6 in are both tripled, then the perimeter is ________ and the area is multiplied by ________. 11. If the base and height of a rectangle with length 18 ft and width 6 ft are both multiplied by ½, then the perimeter is multiplied by _______ and the area is multiplied by _________. tripled 9 or 32 1/2 ¼ or (1/2) 2

Example: 12 Describe the effect on the area for changing the following. The diagonal SU of the kite with vertices R(2, 2), S(4, 0), T(2, –2), and U(–5,0) is multiplied by . original dimensions:

Example: 13 The base and height of the triangle with vertices P(2, 5), Q(2, 1), and R(7, 1) are tripled. Describe the effect on its area and perimeter. original dimensions: The perimeter is tripled, and the area is multiplied by 9. dimensions tripled:

Example: 14 A circle has a circumference of 32 in. If the area is multiplied by 4, what happens to the radius? and the area is A = r2 = 256 in2. If the area is multiplied by 4, the new area is 1024 in2. The original radius is r2 = 1024 Set the new area equal to r2. r2 = 1024 Divide both sides by . Take the square root of both sides and simplify. r = √1024 = 32 Notice that 32 = 2(16). The radius is multiplied by 2, which is the square root of 4.