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 index radical sign radicand The expression under the radical sign is the radicand. The index defines the root to be taken.

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

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

leg a b c hypotenuse 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:

a b c 7.3 – Square Roots and The Pythagorean Theorem a2a2 a c2c2 b2b2 c b The sum of the areas of the two smaller squares is equal to the area of the larger square.

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

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

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

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

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

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 B C  ABC x y z E F D  DEF y x z Triangle ABC is congruent to triangle DEF.  A =  D  B =  E  C =  F AB = DE BC = EF CA = FD  ABC   DEF

7.4 – Congruent and Similar Triangles Determining Congruent Triangles A B C  ABC E F D  DEF by SSS AB = DEBC = EFCA = FD  ABC   DEF 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.

7.4 – Congruent and Similar Triangles Determining Congruent Triangles  DEF A B C  ABC ° by SAS AC = DF  C =  F BC = EF  ABC   DEF 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. E F D °

7.4 – Congruent and Similar Triangles Determining Congruent Triangles by ASA  C =  F BC = EF  ABC   DEF 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. A B C  ABC 12 35° 25°  DEF E F D 12 35° 25°  B =  E

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

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

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

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

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