1. a pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a,b and hypotenuse c and a^2 + b^2 = c^2.a. 3^2 + 4^2 = 9 + 16 = 25 = 5^2 :- 3,4,5 are pythagorean tripleb. 6^2 + 8^2 = 36 + 64 = 100 = 10^2 ≠ 9^2 :- 6,8,9 are not pythagorean triplec. 15^2 + 20^2 = 225 + 400 = 625 = 25^2 :- 15,20,25 are pythagorean tripled. 10^2 + 24^2 = 100 + 576 = 676 = 26^2 :- 10,24,26 are pythagorean tripleso, b. 6,8,9 are not pythagorean triple2. i. by square the two shorter sides and adding them together :4^2 + 5^2 = 16 + 25 = 41ii. by square the longest side :6^2 = 36iii. from i and ii above : 4^2 + 5^2 ≠ 6^2so, 4,5,6 are not pythagorean triple and this do not satisfy the pythagorean theorem.thus, the triangle is not right-angled.

in order for a triangle to be "pythagorean", the sum of the square of the two shorter sides must equal the square of the longest side. so, considering each question individually:1. a)3 x 3 = 94 x 4 = 165 x 5 = 259 + 16 = 25so this is a pythagorean triangle1. b)6 x 6 = 368 x 8 = 649 x 9 = 8136 + 64 does not equal 81this is not a pythagorean triangle1. c)15 x 15 = 22520 x 20 = 40025 x 25 = 625225 + 400 = 625this is a pythagorean triangle.1 d)10 x 10 = 10024 x 24 = 57626 x 26 = 676this is a pythagorean triangle! :)2. i) 4 x 4 = 165 x 5 = 2516 + 25 = 412 ii)6 x 6 = 362 iii)16 + 25 does not equal 36this is not right angled

1) applying the pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse, we can determine which of these sets is not a pythagorean triple (bold font): to. for set 3, 4, 5: 3^2 + 4^2 = 9 + 16 = 25 5^2 = 25 so, 3, 4, 5 is a pythagorean triple. b. for set 6, 8, 9: 6^2 + 8^2 = 36 + 64 = 100 9^2 = 81 these numbers do not satisfy the pythagorean theorem, therefore they do not form a pythagorean triple. c. for the set 15, 20, 25: 15^2 + 20^2 = 225 + 400 = 625 25^2 =625 therefore, these numbers form a pythagorean triple. d. for the set 10,24,26: 10^2 +24^2=100+576=676 26^2=676 the set then forms a pythagorean triple.2) taking into account that the triangle has sides of length 4, 5 and 6: i) if you square the two shorter sides and add them, you get: 4^2 + 5^2 = 16 + 25 = 41 ii) if you square the longest side, you get: 6^2 = 36iii) to determine if the triangle is right angled, we can compare the sum of the squares of the two shorter sides with the square of the longer side. if the sum of the squares of the two shorter sides is equal to the square of the longer side, then the triangle is right angled. in this case: sum of squares of shorter sides = 41 square of longest side = 36 since 41 is not equal to 36, the triangle with sides of length 4, 5, and 6 is not a right triangle.any inquiries at your disposal! thank you so much!!