3.4 – Geometric problems – 5 step approach (p200) Understand problem underline / accent important words, data & question sketches, tables, graphs Translate words into math Choose variable & express other unknowns in terms of variable Express known quantities as algebraic expressions / equation Solve math equation Check answer Answer original question in words using math solution Understand – Variable - Equation Words Math Check - Solve

Equilateral means all three sides are the same length. The perimeter of an equilateral triangle is 44.4 meters. What is the length of each side? Equilateral means all three sides are the same length. Variable: Let x = length of one of the sides. Equation & Solve Perimeter of triangle = sum of 3 sides (known property) 44.4meters = x + x + x (for equilateral triangle) 44.4m = 3x (combined like terms) 44.4 / 3 = x (multiplied both sides by 1 / 3) 14.8 = x Check: 3(14.8) = 44.4 Answer: The length of each side is 14.8meters.

Let a = measure of angle A. Then measure of angle B = a-16 Angles A and B are complementary angles, and angle B is 16 degrees less than angle A. Find the measure of angle A and angle B. Complementary angles sum to 90 degrees. Variable: Let a = measure of angle A. Then measure of angle B = a-16 Equation & Solve sum of angles = 90 (known property) a + (a - 16) = 90 (variable choice) 2a – 16 = 90 (collected like terms) 2a = 90 + 16 (added 16 to both sides) 2a = 106 (collected like terms) a = 53 (multiplied both sides by 1 / 2) Thus angle B = 53 – 16 = 37 Check: 53 + 37 = 90 Answer: The measure of angle A is 53 degrees and the measure of angle B is 37 degrees.

- supplementary angles sum to 180 degrees. Angles A and B are supplementary angles, and angle B is 4 times as large as angle A. Find the measure of angle A and angle B. - supplementary angles sum to 180 degrees. Variable: Let a = measure of angle A. Then measure of angle B = 4a Equation & Solve: sum of angles = 180 a + 4a = 180 5a = 180 (collected like terms) a = 180 / 5 (multiplied both sides by 1 / 5) a = 36 thus angle B = 4(36) = 144 Check: 36 + 144 = 180 Answer: The measure of angle A is 36 degrees and the measure of angle B is 144 degrees.

- vertical (opposite) angles are equal. A pair of vertical (opposite) angles are represented: 3x +30 and 4x + 20. Determine their numerical measures. 3x +30 - vertical (opposite) angles are equal. 4x+20 Variable: Variable = x is given Two angles are 3x +30 and 4x + 20. Equation & Solve: 3x + 30 = 4x + 20 (known property) 30 = 4x – 3x + 20 (add -3x to both sides) 30 = x + 20 (collected like terms) 30 – 20 = x (subtract 20 from both sides) 10 = x (collected like terms) Then 3(10) + 30 = 60 Check: 3(10) + 30 = 60 = 4(10) + 20 = 60 Answer: The measure of the angles is 60 degrees each.

- the angles of a triangle sum to 180 degrees. One angle of a triangle is 15 degrees larger than the smallest angle, and the third angle is 3 times as large as the smallest angle. Find the measures of the three angles. - the angles of a triangle sum to 180 degrees. Variable: Let s = measure of the smallest angle in the triangle. Then the other angles have measures s + 15 and 3s. Equation & Solve: s + (s + 15) + 3s = 180 (known property for triangles) 5s + 15 = 180 (collected like terms) 5s = 180 - 15 (subtract 15 from both sides) 5s = 165 (collected like terms) s = 165 / 5 (multiplied both sides by 1 / 5) s = 33 then s + 15 = 48 and 3s = 99. Check: 33 + 48 + 99 = 180 ; 33 + 15 = 48 ; 3(33) = 99 Answer: The measure of the angles is 33, 48 and 99 degrees.

- Perimeter = 2length + 2 width Sue is planning to build a rectangular sandbox for her niece. She wants the box length to be twice the width, and she has 12m of lumber to build the sides with. What dimensions should she use? - Perimeter = 2length + 2 width Variable: Let w = width of sandbox in meters Then the length is 2w. Equation & Solve: 12 = 2(2w) + 2w (known formula, with our variables) 12 = 4w + 2w (remove parentheses) 12 = 6w (collected like terms) 12 / 6 = w (multiplied both sides by 1 / 6) 2 = w then length = 2(2) = 4 Check: 2(4) + 2(2) = 12 and 4 = 2(2) Answer: The sandbox needs to be 2meters by 4 meters.

Let w = width of bookcase in cm A bookcase is to have 4 shelves. The height of the bookcase is to be 20cm greater than the width; only 7m of lumber is available. Ignore top and back of case. What should the width and height be? lumber = 2 sides + 4 shelves h Variable: Let w = width of bookcase in cm Then the height is 20 + w. w Equation & Solve: 700cm = 2(20 + w) + 4w (our formula from sketch) 700 = 40 + 2w + 4w (distribute) 700 = 40 + 6w (collected like terms) 660 = 6w (added -40 to both sides) 660 / 6 = w (multiplied both sides by 1 / 6) w = 110cm and then height = 110 + 20 = 130cm Check: 2(130) + 4(110) = 260 + 440 = 700 Answer: The bookshelf needs to be 130cm tall and 110cm wide..

Let L = length of single paddock in meters Joe needs to divide his farm into 4 equal paddocks along the banks of a river. The total amount of fencing available is 1804m. Each paddock needs to have the width as 80% of the length. What are the dimensions of each paddock, and the dimensions of the total fenced off land? river L Variable: Let L = length of single paddock in meters Then the width of each paddock is 0.8L Equation & Solve: 1804m = 5L + 4(0.8L) (from sketch, with our variables) 1804 = 5L + 3.2L (remove parentheses) 1804 = 8.2L (collected like terms) 1804 / 8.2 = L (multiplied both sides by 1 / 8.2) 220 = L then width = 0.8(220) = 176 Check: 5(220) + 4(176) = 1100 + 704 = 1804 Answer: Each paddock is 220m by 176m; total dimensions: 220m by 704m

3.4 – Geometric problems – 5 step approach (p200) Understand problem underline / accent important words, data & question sketches, tables, graphs Translate words into math Choose variable & express other unknowns in terms of variable Express known quantities as algebraic expressions / equation Solve math equation Check answer Answer original question in words using math solution Understand – Variable - Equation Words Math Check - Solve