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1 PROPORTIONS REVIEW ALTITUDE FROM HYPOTENUSE FORMS SIMILAR TRIANGLES PROBLEM 1a PROBLEM 1b PROBLEM 2 PROBLEM 3 PROBLEM 4 PROBLEM 5 PROBLEM 6 STANDARDS 4, 5, 14, 15 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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2 Standard 4: Students prove basic theorems involving congruence and similarity. Los estudiantes prueban teoremas básicos de congruencia y semejanza. Standard 5: Students prove triangles are congruent or similar and are able to use the concept of corresponding parts of congruent triangles. Los estudiantes prueban que los triangulos son congruentes o semejantes y son capaces de usar el concepto de partes correspondientes en triángulos congruentes. Standard 14: Students prove the Pythagoream Theorem. Los estudiantes prueban el Teorema de Pitágoras. Standard 15: Students use the Pythagoream Theorem to determine distance and find missing lenghts of sides of right triangles. Los estudiantes usan el Teorema de Pitágoras para determinar distancia y el lado faltante de un triángulo rectangulo. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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3 A D A and D are the EXTREMES B C B and C are the MEANS Cross-multiplying: (A)(D)=(C)(B)The product of the MEANS is equal to the product of the EXTREMES PROPORTIONS REVIEW = A D B C = A D B C = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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4 x x 10 40 = Find the GEOMETRIC MEAN of 10 and 40: x x 10 40 Cross-multiplying (10)(40)=(x)(x) Applying symmetric property x = 400 2 2 |x|=20x=20 and x=-20 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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5 x x6 24 = Find the GEOMETRIC MEAN of 6 and 24: x x6 24 Cross-multiplying (6)(24)=(x)(x) Applying symmetric property x = 144 2 2 |x|=12x=12 and x=-12 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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6 D A B C How many similar triangles Altitud CD in ABC form? STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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7 A D B C A C B A D C C DB STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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8 AD AC A B A D C CB DC STANDARDS 4, 5, 14, 15 C PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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9 A CB A D C AC AD AB AC STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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10 A C B A D C CB DC AB AC STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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11 A D C C DB AD DC DB AD DC AC CB AC CB DC DB STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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12 A C B C DB AC CD CB DB AC CD AB CB AB CB DB STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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13 A C B A D C C DB AC AD CB DC AC AD AB AC AB AC CB DC AD CD DC DB AD CD AC CB AC CB DC DB AC CD CB DB AC CD AB CB AB CB DB STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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14 A D B C A C A D C DB C B AC CD CB DB AC CD AB CB AB CB DB ACB CDB AD CD DB AD CD AC CB AC CB DC DB ADC CDB AC AD CB DC AC AD AB AC AB AC CB DC ACB ADC STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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15 z y x 17 8 1. STANDARDS 4, 5, 14, 15 Find the value for x, y and z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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16 z y x 17 8 z z 8 8 x y x+y 1. STANDARDS 4, 5, 14, 15 Find the value for x, y and z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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17 8 x x + 8 = 17 22 2 -64 x + 64 = 289 2 x = 225 2 2 |x|=15 x=15 and x=-15 1. STANDARDS 4, 5, 14, 15 Find the value for x, y and z: z y x 17 8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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18 8 8 17 x= 15z y (17) z = (8)(17) 15 z 17 8 15 z = 9.0. 1. STANDARDS 4, 5, 14, 15 Find the value for x, y and z: = z 17 8 15 = z y x 17 8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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19 8 8 17 x= 15z y y 8 8 15 (8) y = (8)(8) 15 1. STANDARDS 4, 5, 14, 15 Find the value for x, y and z: = y 8 8 15 = y = 4.3. z y x 17 8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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20 10 x 5 y z 2. STANDARDS 4, 5, 14, 15 Find the value for x, y and z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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21 2. zz x x y y 5 10+5 10 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: 10 x 5 y z PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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22 x 10+5 10 z z y 5 x y 2. Cross-multiplying (y)(y)=(10)(5) y = 50 2 2 |y|=7 y =7 and y=-7 5 y y 10 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: = 5 y y 10 = x 5 y z PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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23 x 10+5 10z y=7 5 x 2. z 10 + 7 = z 2 2 2 z = 149 2 2 100 + 49 = z 2 |z|=12.2 z =12.2 and z=-12.2 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: 10 x 5 y z PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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24 x 10+5 10 z y=7 5 x 2. z 5 + 7 = x 22 2 x = 74 2 2 25 + 49 = x 2 |x|=8.6 x=8.6 and x = -8.6 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: 10 x 5 y z PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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25 x The length of a rectangle is 6 in and its width is 4 in. Find the length of its diagonal. 6 in 4 in 6 + 4 = x 22 2 x = 52 2 36 + 16 = x 2 x = 52 2 |x| = 7.2 x = 7.2 and x = -7.2 3. STANDARDS 4, 5, 14, 15 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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26 z y 22 26 x 4. STANDARDS 4, 5, 14, 15 Find the value for x, y and z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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27 z y 22 26 x 4. z+22 z xx 22 y y 26 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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28 4. y z+22 z x x 26 22 y 26 (26) z = (26)(26) 22 z 26 22 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: = z 26 22 = z = 30.72 z y 22 26 x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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29 4. y z+22 x 26 22 y 26 xz=30.7 30.7 + 26 = x 22 2 x = 1618.5 2 942.5 + 676 = x 2 x = 1618.5 2 |x| =40.2 x= 40.2 and x =-40.2 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: z y 22 26 x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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30 4. y z+22 yxx 26 22 y 26 y = 1160 2 2 676 + 484 = y 2 |y|=34 y=34 and y =-34 26 + 22 = y 2 2 2 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: z y 22 26 x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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31 5. z y x 16 9 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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32 z y x 16 9 5. y y zz x x 9 9+16 16 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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33 16 x 9 x = 5. y 9+16 16zz x 9 y x x = 144 2 2 x 9 x |x|=12 x=12 and x=-12 Cross-multiplying (x)(x)=(16)(9) STANDARDS 4, 5, 14, 15 Find the value for x, y and z: = z y x 16 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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34 5. y 16z z x=12 9 y z = 400 2 2 256 + 144 = z 2 |z|=20 z =20 and z=-20 16 + 12 = z 2 2 2 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: z y x 16 9 9+16 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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35 5. y 16 z z x=12 9 y y = 225 2 2 81 + 144 = y 2 9 + 12 = y 2 2 2 |y|=15 y=15 and y=-15 STANDARDS 4, 5, 14, 15 Find the value for x, y and z: z y x 16 9 9+16 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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