J OURNAL C HAPTER 7- 8 Marcela Janssen

C HAPTER 7: S IMILARITY

R ATIOS AND P ROPORTIONS. What is a Ratio? Ratio: a comparison of two quantities by division. A ratio can be written as: - a to b - a:b - a/b What is a proportion? Proportion: A statement were two ratios are equal. Ex. a/b = c/d

Do proportions and ratios have any relationship? If it does explain it. A proportion: 2 equal ratios How can a proportion be solved? To solve a proportion you need to cross-multiply. How can I check a proportion is equal? After finding the value of any variables, substitute each variable into the original proportion. Then, cross- multipe and make sure both sides are equal.

E XAMPLE 1 What is the ratio that expresses the slope of U ? Slope = rise run 3 – (-2) = _5 = 1 5 – (-5) 10 2

E XAMPLE 2 Solve the proportion. 10 = 90 y (126) = y(90) 1260 = 90y 1260 = y = y Check: 10 = (126) = 14(90) 1260 = 1260 Correct

E XAMPLE 3 2 = x – 3 4 (X-3) 50 4 (x-3) 2 = 2(50) 4 (x-3) 2 = 100 (x-3) 2 = 25 x-3 = + 5 x-3=5 x=8 x-3 = -5 x=2 x = 8 or 2 Check: 2 = (8-3) 50 2(50) = 4 (8-3) = 4 (5) = 4 (25) 100 = 100

S IMILAR P OLYGONS AND S CALE F ACTOR How do i know two poygons are similar? For two polygons to be similar they must have the same shape but different measurements. What is a Scale Factor? Scale factor is the multiplier used on each dimension to change one figure into a similar figure.

E XAMPLE 1 Determine whether ABC and DEF are similar. If so, write the similirity ratio and a similirity statement.

It is given that A = D and B = E. C = F by the Third angle theorem. AB = BC = AC = 2. Thus the DE EF DF 3 similarity ratio is 2, and ABC is similar 3 to DEF.

S CALE FACTOR FOR PERIMETERS AND AREAS

S IMILAR T RIANGLES AND I NDIRECT MEASUREMENTS. Similar tiangles are triangles that have the same angles and have a ratio. Indirect measurement uses formulas, similar figures and/or proportions. How can similar triangles can be used to make an indirect measurement? By using similar triangles you can find the missing length of one of the sides of one of the triangles with proportions.

E XAMPLE 1

E XAMPLE 2 – R EAL LIFE EXAMPLE Peter wants to know how high the cliff is. He is 10 m away from the cliff and his eyes visual field watching directly straight is 5 m from the floor. How high is tha cliff? 10 = x_ x = 100 X= 20 Answer: the cliff is 20 m high.

E XAMPLE 3 Real Life Situation. Jenny´s cat went all the way up the tree´s top. She doesn´t know how tall the tree is. Her ladder is 120 m and she wants to know if she needs to borrow the ladder of his neighbor. The shadow the tree makes is 10.2 m and her shadow is 0.8 m. Jenny´s height is 12 m. Does Jenny need to borrow the ladder?

x = x = 10.2(12) 0.8x = x = X = 153 m Answer: Yes, Jenny needs to borrow a ladder because hers is too short.

R IGHT T RIANGLE A LTITUDE P ROPORTIONALITY T HEOREM The altitude to the hypothenuse of a right triangle that are similar to each other and to the original triangle. Proportions can be used to solve real life problems such a tree height or to know the distance from one side to the other in a river.

E XAMPLES For each example write a similarity statement. WX = WY WY WZ

CE = CD BC CE PS = PQ SR PS

C HAPTER 8: R IGHT T RIANGLES AND T RIGONOMETRY

TRIGONOMETRIC RATIOS Sine: opposite leg hypothenuse Cosine: adjacent leg hypothenuse Tangine: opposite leg adjacent leg

WAYS TO REMEMBER THE TRIGONOMETRIC RATIOS SOH CAH TOA Sin: opposite leg hypothenuse Cos: adjacent leg hypothenuse Tan: opposite leg adjacent leg

E XAMPLE 1 Sin A = 3 5 Cos A = 4 5 Tan A = 3 4

E XAMPLE 2 Use Sin, Cos,Tan or Pytagorean Theorem to find x and y. Cos 52= y 6 6 Cos 52 = y y = 3.7 Sin 52 = x 6 6 Sin 52= x X= 4.72

E XAMPLE 3 Find the length of AB and round to the nearest hundred. Tan 34= 4.2 AB AB tan 34 = 4.2 AB = 4.2 tan 34 AB = 6.23 in

A NGLES OF E LEVATION AND D EPRESSION Angle of elevation: the angle formed by a horizontal line and a line connecting to a point above the horizontal line. Angle of depression: the angle formed by a horizontal line and a line connecting to a point below.

E XAMPLES Angle of Deppression: The line is going down. Angle os elevation: The line is going upwards.

R UBRIC _ YESS _(0-10 pts) Describe a ratio. Describe a proportion. How are they related? Describe how to solve a proportion. Describe how to check if a proportion is equal. Give 3 examples of each. __ NOO _(0-10 pts) Describe what it means for two polygons to be similar. What is a scale factor? Give at least 3 examples of each. ___NOOO__(0-10 pts) Describe how to find the scale factor for the perimeter and areas of similar figures. Give at least 3 examples of each one. _ YESSS _(0-10 pts) Describe how to use similar triangles to make an indirect measurement. Give at least 3 examples. __NOOO___(0-10 pts.) Describe the right triangle altitude proportionality theorem. Give at least 3 examples. Explain how the proportions can be used to solve real life problems. __NOOO___(0-10 pts.) Describe the three trigonometric ratios. Explain how they can be used to solve a right triangle. What does it mean to solve a triangle? Give at least 3 examples of each. How are they used in real life? ___NOOO__(0-10 pts.) Compare an angle of elevation with an angle of depression. How are each used? Give at least 3 examples of each. _____(0-5pts) Neatness and originality bonus _____Total points earned (80 possible)

M R. T URNER : Please check Ratio and proportion Similar triangles and scale factor trigonometric ratios