Solving Proportions: Find x: 4 x = x. 2 = x x = 3x. 5 = 3x x = x 3 = 3 x. x =
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1 Similar Triangles Page 1 Solving Proportions: Find x: 1. 4 x = x 2 = x + 4 x = 3x 2x 4. 7 x 5 = 3x x = x 3 = 3 x 7. x = x = ircle the true proportions:. 6 8 = = = 14 21
2 Similar Triangles Page 2 Find x (not all of the following are proportions): 1. x 3 + 5x 6 =1 2. x + 2 x + 3 = x +1 4x 1 = x 7 x 3 = x 5 + x 6 = 1 2x 1 6. = x x 5 = x x 8 = 2 x 9. 4x x 6 4 = 2x x = 2x 1 3
3 Similar Triangles Page 3 1. Reduce the following ratio: 2. Reduce the following ratio: x 2 3x 40 x 2 + 6x x 2 96x x 2 80x Solve for x: 4. Solve for x: 5. Solve for x: x +1 x + 3 = 2 x 2x 2 = 3x 5 3x 3 x 9 = 3 x 6. Solve for x and y: 7. Solve for x and y: 3x + 2 = 5y y 6x = y 2x = x +11 = y Find the ratio of y to x and find the ratio of x to y if 3y ax = 2x + by. 9. Find x:y and y:x if 3 x + y = 5 y x 10. Find x:y and y:x if 6 x + 3 = 12 y If 10y=4x, find x:y 12. If ax cy=dy+bx find y x
4 Similar Triangles Page Show that a b = c d is equivalent to a c = a + b c + d 14. (ST: May 1997) 15. (ST: November 1996) If 10 a = b 12, what is the value of ab? If x y = 1, then x +y = 16. (ST: March 1994) In 1980, the ratio of male students to female students at Frost ollege was 2 males to 3 females. Since then, the enrollment of male students in the college has increased by 400 and the enrollment of female students has remained the same. The ratio of males to females is currently 1 to 1. How many students are currently enrolled at Frost ollege? 17. (ST: May 1996) If the ratio of q to r is 4 to 5, which of the following could be true? a. q = 0, r = 4 5 b. q = 2, r = 5 c. q = 5, r = 6 2 d. q = 15, r = 12 e. q = 16, r = (ST: January 1997) If x = 2y, y = 4z, 2z = w, and w 0, then what is x w?
5 Similar Triangles Page 5 1. JKL ~ Similar Triangles. Which angles are equal?. Which sides are proportional? 2. R ~ GL Mark the equal angles R 7 G L. Find L. Find GL 3. WN ~ T Mark the equal angles. N W 12 T. Find T.. Find T
6 Similar Triangles Page Find UK.. Find N. 5. = and = F. Therefore, ~ by theorem. Find x. = 2x 18, F = 4, = 5, = x F
7 Similar Triangles Page 7 6. ~ F Mark the equal angles. Find x. = 3, F = 4, = x, = 68 F 7. RG ~ UP U 2x+2 4x 2 R 6 G 9 P. Find R.. Find U
8 Similar Triangles Page 8 8) a. In the diagram below, Δ : Δ b. Find c. Find the coordinates of ( 9, 44) (1, 20) (0, 0) (6, 8) 9) Find x and y. 5
9 Similar Triangles Page 9 1. When a line is parallel to one side of a triangle it:. uts the sides proportionally.. Forms a small triangle similar to the large one. omplete the following proportions:. =. =. =. =. = F. = 2. Solve for x: = 16, = x, = 4, = 3
10 Similar Triangles Page Solve for x: = 3, = 4, = x + 5, = x + 10, 4. Solve for x: = 2, = 3, = x + 4, = x + 10,. 5. Solve for x: = 5, = 2, = x + 6, = x 6
11 Similar Triangles Page Solve for x: = 3, = 4, = x 6, = x + 6,. 7. Solve for x: = 5, = 2, = 3x + 6, = 5x Find the value of x: x
12 Similar Triangles Page 12 The Similarity Theorem For any two triangles, if two pairs of corresponding angles are equal, then the triangles are similar. PQR ~ R Q P 1. Which pairs of triangles are similar by the Theorem?
13 Similar Triangles Page 13 Using the Theorem to find the lengths of sides of triangles. 1. In RST, GH RS, RS = 18, GH = 6 and GT = 9. S G R H T. = ecause:. = ecause:. ~ by the Theorem. Find ST. 2. In TR and MSR, T SM, T = 21, SM = 14 and TR = 18. T S R M. = ecause:. = ecause:. ~ by the Theorem. Find MR.
14 Similar Triangles Page Which pairs of triangles are similar by the Theorem? rod 3 meters tall casts a shadow 5 meters long. t the same time, the shadow of a tower is 110 meters long. The sun s rays form equal angles with both the tower and the rod. P x 3 R 110 Q 5 How tall is the tower?
15 Similar Triangles Page Q Two olumn Statement Reason Proofs with Similar Triangles R T Given: PQ ST Prove; PQR ~ TSR S P Statement Reason 1. PQ ST Q = S P = T PQR ~ TSR X O Given: XY ~ O Prove: O ~ O Y Statement Reason 1. XY ~ O = O = O3. 4. O ~ O 4.
16 Similar Triangles Page Given: bisects = Prove: ~ Statement Reason Given: = F = F Prove: F ~ F F Statement Reason
17 Similar Triangles Page 17 pplying ratios and proportions to word problems 1. Susan and ebbie were building a model plane with a scale of 2:143. a. When they were done, they measured the model and found that it was 23 in long. They then calculated the length of the real plane and found that it was ft long. b. Susan and ebbie lost their ruler and they could not measure their model s wingspan. The box that the model had come in said that the real plane had a wingspan of 49 ft, so Susan and ebbie figured out that their model had to have a wingspan of inches. 2. The ratio of males to females in a room is 2 to 3. If there are 625 people total in the room, how many males are there? 3. dog and a puppy together weigh 56 pounds with the ratio of their weights being 5 to 2. How much does the puppy weigh? 4. The sides a, b and c of a triangle correspond, respectively to the sides r, s and t of a similar triangle. Find the length of side t in terms of b, c and s. 5. Δ : ΔXYZ with =5, =7, and =10. The perimeter of ΔXYZ is 33. Find the length of the longest side of ΔXYZ.
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