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7.3 – Square Roots and The Pythagorean Theorem

7.3 – Square Roots and The Pythagorean Theorem. Finding a square root of a number is the inverse operation of squaring a number. This symbol is the radical or the radical sign. radical sign. index. radicand. The expression under the radical sign is the radicand.

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7.3 – Square Roots and The Pythagorean Theorem

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  1. 7.3 – Square Roots and The Pythagorean Theorem Finding a square root of a number is the inverse operation of squaring a number. This symbol is the radical or the radical sign radical sign index radicand The expression under the radical sign is the radicand. The index defines the root to be taken.

  2. 7.3 – Square Roots and The Pythagorean Theorem The above symbol represents the positive or principal square root of a number. This symbol represents the negative square root of a number.

  3. 7.3 – Square Roots and The Pythagorean Theorem A square root of any positive number has two roots – one is positive and the other is negative. If a is a positive number, then is the positive square root of a and is the negative square root of a. Examples: non-real #

  4. 7.3 – Square Roots and The Pythagorean Theorem The Pythagorean Theorem: A formula that relates the lengths of the two shortest sides (legs) of a right triangle to the length of the longest side (hypotenuse). The Pythagorean Theorem: hypotenuse c leg a b leg

  5. 7.3 – Square Roots and The Pythagorean Theorem The sum of the areas of the two smaller squares is equal to the area of the larger square. c2 c c a2 a b a b2 b

  6. 7.3 – Square Roots and The Pythagorean Theorem Find the length of the hypotenuse of the given right triangle. The Pythagorean Theorem: hypotenuse c 12 feet a b 16 feet

  7. 7.3 – Square Roots and The Pythagorean Theorem Find the length of the hypotenuse of the given right triangle. The Pythagorean Theorem: hypotenuse c 5 feet a b 12 feet

  8. 7.3 – Square Roots and The Pythagorean Theorem Find the length of the leg of the given right triangle. The Pythagorean Theorem: 10 meters c 6 meters a b

  9. 7.3 – Square Roots and The Pythagorean Theorem Find the length of the hypotenuse of the given right triangle. The Pythagorean Theorem: hypotenuse c 4 feet a b 7 feet Use the Square Root Table

  10. 7.3 – Square Roots and The Pythagorean Theorem Find the length of the missing side of the given right triangle. 11 inches 14 inches Use the Square Root Table The Pythagorean Theorem:

  11. 7.4 – Congruent and Similar Triangles Congruent Triangles: Triangles that have the same shape and size. The measures of the corresponding angles and sides are equal. A D x y x y z z C B F E DEF ABC A = D B = E C = F BC = EF CA = FD AB = DE Triangle ABC is congruent to triangle DEF.  ABC   DEF

  12. 7.4 – Congruent and Similar Triangles Determining Congruent Triangles Side–Side–Side (SSS): If the lengths of the three sides of a triangle are congruent (equal) to the corresponding sides of another triangle, then the triangles are congruent. F A 16 12 7 7 C E D B 16 12 ABC DEF BC = EF CA = FD AB = DE by SSS  ABC   DEF

  13. 7.4 – Congruent and Similar Triangles Determining Congruent Triangles Side–Angle–Side (SAS): If the lengths of the two sides and the angle between them of a triangle are congruent (equal) to the corresponding sides and the angle between them of another triangle, then the triangles are congruent. D A 5 5 40° 40° E C B F 8 8 ABC DEF BC = EF AC = DF C = F by SAS  ABC   DEF

  14. 7.4 – Congruent and Similar Triangles Determining Congruent Triangles Angle–Side–Angle (ASA): If the measures of the two angles and the side between them of a triangle are congruent (equal) to the corresponding angles and the side between them of another triangle, then the triangles are congruent. D A 35° 25° 25° 35° E C B F 12 12 ABC DEF BC = EF B = E C = F by ASA  ABC   DEF

  15. 7.4 – Congruent and Similar Triangles Determining Congruent Triangles Are the following pairs of triangles congruent? State the reason. 42 42 Q M 28° 28° R O N S 35 35 N = R MNO QRS MN = QR NO = RS by SAS  MNO   QRS

  16. 7.4 – Congruent and Similar Triangles Determining Congruent Triangles Are the following pairs of triangles congruent? State the reason. 15 J X L = Z 15 37° 37° Y L K Z  JKL   XYZ 29 L J = ZX KL  YZ 28

  17. 7.4 – Congruent and Similar Triangles Determining Congruent Triangles Are the following pairs of triangles congruent? State the reason. R 26° E P P = A E = G 10 8 A by ASA L PE = GA  PRE   ALG 10 26° G

  18. 7.4 – Congruent and Similar Triangles Similar Triangles Similar Triangles: Triangles whose corresponding angles are equal and the corresponding sides are proportional. Triangle RIP is similar to triangle AXE.  RIP   AXE R A I X E 10 8 P The ratio of the corresponding sides is: AE AX 8 EX = = = RP 10 RI PI

  19. 7.4 – Congruent and Similar Triangles  ABC   DEF Triangle ABC is similar to triangle DEF. Find the values of x and y. A D 24 y x 15 E C F B 30 36 6 BC 36 The ratio of the corresponding sides is: = = 5 EF 30 6 24 6 y = = x 5 15 5 = 120 6x = 90 5y x = 20 x = 18

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