Which of the following sets of numbers could represent the three sides of a right triangle?
A{49,64,80\}{49,64,80}
B{45,60,75\}{45,60,75}
C{48,55,72\}{48,55,72}
D{34,56,65\}{34,56,65}
just check to be sure that a^2 + b^2 = c^2
where c is the longest side.
It might help to remember a few basic Pythagorean triples:
3-4-5
5-12-13
etc. Those and any of their multiples will form a right triangle.
The useless duplication is just annoying.
To determine whether a set of numbers could represent the sides of a right triangle, we can use the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the longest side (called the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides.
Let's check each set of numbers using the Pythagorean theorem:
A: {49, 64, 80}
To determine if these numbers form a right triangle, we need to check if 80^2 = 49^2 + 64^2 holds true.
80^2 = 6400
49^2 = 2401
64^2 = 4096
2401 + 4096 = 6497
Since 6497 is not equal to 6400, the set {49, 64, 80} does not represent the sides of a right triangle.
B: {45, 60, 75}
For this set, we need to check if 75^2 = 45^2 + 60^2 holds true.
75^2 = 5625
45^2 = 2025
60^2 = 3600
2025 + 3600 = 5625
Since 5625 is equal to 5625, the set {45, 60, 75} represents the sides of a right triangle.
C: {48, 55, 72}
To check if these numbers form a right triangle, we compare 72^2 with the sum of the squares of the other two sides.
72^2 = 5184
48^2 = 2304
55^2 = 3025
2304 + 3025 = 5329
Since 5329 is not equal to 5184, the set {48, 55, 72} does not represent the sides of a right triangle.
D: {34, 56, 65}
For this set, we compare 65^2 to the sum of the squares of the other two sides.
65^2 = 4225
34^2 = 1156
56^2 = 3136
1156 + 3136 = 4292
Since 4292 is not equal to 4225, the set {34, 56, 65} does not represent the sides of a right triangle.
In conclusion, the only set that represents the sides of a right triangle is B: {45, 60, 75}.
To determine if a set of numbers can represent the sides of a right triangle, we can check if they satisfy the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's calculate the values for each option:
A: {49, 64, 80}
Hypotenuse^2 = 80^2 = 6400
Sum of squares of other two sides = 49^2 + 64^2 = 2401 + 4096 = 6497
B: {45, 60, 75}
Hypotenuse^2 = 75^2 = 5625
Sum of squares of other two sides = 45^2 + 60^2 = 2025 + 3600 = 5625
C: {48, 55, 72}
Hypotenuse^2 = 72^2 = 5184
Sum of squares of other two sides = 48^2 + 55^2 = 2304 + 3025 = 5329
D: {34, 56, 65}
Hypotenuse^2 = 65^2 = 4225
Sum of squares of other two sides = 34^2 + 56^2 = 1156 + 3136 = 4292
Now we can compare the values:
A: Hypotenuse^2 ≠ Sum of squares of other two sides
B: Hypotenuse^2 = Sum of squares of other two sides
C: Hypotenuse^2 ≠ Sum of squares of other two sides
D: Hypotenuse^2 ≠ Sum of squares of other two sides
Based on our calculations, option B {45, 60, 75} is the only set that could represent the three sides of a right triangle, as it satisfies the Pythagorean theorem.