WarmUp Exercises 1 Solve 12 x 10 60

Warm-Up Exercises 1. Solve 12 = x 10 60 ANSWER 72 2. The scale of a map is 1 cm : 10 mi. The actual distance between two towns is 4. 3 miles. Find the length on the map. ANSWER 0. 43 cm 3. A model train engine is 9 centimeters long. The actual engine is 18 meters long. What is the scale of the model? ANSWER 1 cm : 2 m

6. 1 Similar Polygons 1. Ratio – a comparison of two quantities using the same unit. 2. A ratio can be written in 3 ways a: b a to b 3. Proportion – 2 equal ratios a and d are called extremes b and c are called means

Examples Simplify 1. 35 m 700 m 2. 12 ft 5 yd 3. D 1/20 12 = 4 15 5 F E DF = 45 DE = 15 EF = 30 x + 2 x = 45 3 x = 45 x = 15

6. 1 Similar Polygons • Similar figures have the same shape but are different in size. • In Similar polygons – corresponding angles are congruent and corresponding side lengths are proportional. B C A D F G E H

EXAMPLE Warm-Up 1 Exercises Use similarity statements In the diagram, ∆RST ~ ∆XYZ a. List all pairs of congruent angles. b. Check that the ratios of corresponding side lengths are equal. c. Write the ratios of the corresponding side lengths in a statement of proportionality.

EXAMPLE Warm-Up 1 Exercises Use similarity statements SOLUTION a. b. c. ~ R= X, ~ S= Y and T~ = Z 5 ST 30 25 TR 5 RS 5 = 18 = 3 ; ZX = 15 = 3 = 20 = 3 ; XY 12 YZ Because the ratios in part (b) are equal, RS = ST = TR. YZ ZX XY

Warm-Up Exercises GUIDED PRACTICE 1. for Example 1 Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality. ANSWER ~ J= P, ~ K= Q and L~ = R ; JK = KL LJ = PQ QR RP

6. 1 continued • Scale factor – ratio of the lengths of any 2 corresponding sides in similar polygons. R T x 20 10 S P 16 15 Q y U

EXAMPLE Warm-Up 2 Exercises Find the scale factor Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ZYXW to FGHJ.

EXAMPLE Warm-Up 2 Exercises Find the scale factor SOLUTION STEP 1 Identify pairs of congruent angles. From the diagram, you can see that Z F, Y G, and X H. Angles W and J are right angles, so W J. So, the corresponding angles are congruent.

EXAMPLE Warm-Up 2 Exercises Find the scale factor SOLUTION STEP 2 Show that corresponding side lengths are proportional. ZY = FG 25 20 XW = HJ 15 12 = 5 4 YX = GH 30 24 = 5 4 WZ = JF 5 20 = 4 16 = 5 4

EXAMPLE Warm-Up 2 Exercises Find the scale factor SOLUTION The ratios are equal, so the corresponding side lengths are proportional. ANSWER So ZYXW ~ FGHJ. The scale factor of ZYXW to FGHJ is 5. 4

EXAMPLE Warm-Up 3 Exercises Use similar polygons ALGEBRA In the diagram, ∆DEF ~ ∆MNP. Find the value of x.

EXAMPLE Warm-Up 3 Exercises Use similar polygons SOLUTION The triangles are similar, so the corresponding side lengths are proportional. MN = NP EF DE 12 = 20 x 9 12 x = 180 x = 15 Write proportion. Substitute. Cross Products Property Solve for x.

Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 In the diagram, ABCD ~ QRST. 2. What is the scale factor of QRST to ABCD ? ANSWER 3. 1 2 Find the value of x. ANSWER 8

The polygons are similar. Find x and y. X = 16 Y=1

EXAMPLE Warm-Up 4 Exercises Find perimeters of similar figures Swimming A town is building a new swimming pool. An Olympic pool is rectangular with length 50 meters and width 25 meters. The new pool will be similar in shape, but only 40 meters long. a. Find the scale factor of the new pool to an Olympic pool. b. Find the perimeter of an Olympic pool and the new pool.

EXAMPLE Warm-Up 4 Exercises Find perimeters of similar figures SOLUTION a. Because the new pool will be similar to an Olympic pool, the scale factor is the ratio of the lengths, 4 40 = 5 50 b. The perimeter of an Olympic pool is 2(50) + 2(25) = 150 meters. You can use Theorem 6. 1 to find the perimeter x of the new pool. x = 4 150 5 x = 120 ANSWER The perimeter of the new pool is 120 meters.

Warm-Up Exercises GUIDED PRACTICE for Example 4 In the diagram, ABCDE ~ FGHJK. 4. Find the scale factor of FGHJK to ABCDE. ANSWER 3 2 Find the value of x. ANSWER 12 ANSWER 46 5. 6. Find The perimeter of ABCDE.

Corresponding lengths in similar triangles include side lengths, altitudes, medians, midsegments and so on.

EXAMPLE Warm-Up 5 Exercises Use a scale factor In the diagram, ∆TPR ~ ∆XPZ. Find the length of the altitude PS. SOLUTION First, find the scale factor of ∆TPR to ∆XPZ. TR = 6 + 6 = 12 = 8+8 16 XZ 3 4

EXAMPLE Warm-Up 5 Exercises Use a scale factor Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. 3 PS = Write proportion. 4 PY 3 4 PS 20 = PS = 15 Substitute 20 for PY. Multiply each side by 20 and simplify. ANSWER The length of the altitude PS is 15.

Warm-Up Exercises GUIDED PRACTICE for Example 5 7. In the diagram, ∆JKL ~ ∆ EFG. Find the length of the median KM. ANSWER 42

Warm-Up Exercises 1. Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of EFGH to KLMN. ANSWER Yes; EFGH ~ KLMN; the scale factor is 2: 1

Daily Homework Quiz Warm-Up Exercises 2. In the diagram, of x. ANSWER 13. 5 DEF ~ HJK. Find the value

Warm-Up Exercises 3. Two similar triangles have the scale factor 5 : 4. Find the ratio of their corresponding altitudes and median. ANSWER 4. 5: 4; 5: 4 Two similar triangles have the scale factor 3 : 7. Find the ratio of their corresponding perimeters and areas. ANSWER 3 : 7; 9: 49

Warm Up 1. A man stood so his shadow ended at the same point as the shadow of a monument. Use the information in the figure to find the height of the monument. X ft 6 ft 5. 1 ft 17 ft