Which of the following sets of numbers could represent the three sides of a right triangle?
\{49,64,80\}{49,64,80}
\{45,60,75\}{45,60,75}
\{48,55,72\}{48,55,72}
\{34,56,65\}{34,56,65}
ratio
In right triangle:
a² + b² = c²
49 , 64 , 80
80² = 6400
49² + 64² = 2401 + 4096 = 6497 ≠ 80²
45 , 60 ,75
75² = 5625
45² + 60² = 2025 + 3600 = 5625 = 75²
48 , 55 , 72
72² = 5184
48² + 55² = 2304 + 3025 = 5329 ≠ 72²
34 ,56 , 65
65² = 4225
34² + 56² = 1156 + 3136 = 4292 ≠ 65²
So 45 , 60 ,75 are sides of a right triangle.
Remark:
≠ means different from
Which of the following sets of numbers could not represent the three sides of a right triangle? (57,76,95) ( 10,25,26) (65,72,97) (18,80,82)
Well, as a bot who gets a kick out of triangles, let me share a joke with you first. Why was the math book sad? Because it had too many problems! Now, let's get back to the question.
To determine if a set of numbers 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 hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Now, let's check the options:
1. \{49,64,80\}{49,64,80}: To find out if it forms a right triangle, let's apply the theorem: 49^2 + 64^2 = 80^2. Surprisingly, it fails! So, this one is not a right triangle.
2. \{45,60,75\}{45,60,75}: Let's plug these numbers in the theorem: 45^2 + 60^2 = 75^2. Hooray! It works! This set could represent the sides of a right triangle.
3. \{48,55,72\}{48,55,72}: Now let's apply the theorem: 48^2 + 55^2 = 72^2. Oh no! Another failure. This set does not form a right triangle.
4. \{34,56,65\}{34,56,65}: Applying the theorem: 34^2 + 56^2 = 65^2. It's a jackpot! This set forms a right triangle.
So, the sets \{45,60,75\}{45,60,75} and \{34,56,65\}{34,56,65} represent the three sides of right triangles. As for the others, well, they can still have their own shape and charm, just not a right triangle!
To determine whether a set of numbers could represent the sides of a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's apply this theorem to each set of numbers one by one:
1. Set {49, 64, 80}:
To check if these numbers form a right triangle, we need to see if the Pythagorean theorem holds true.
The largest number, 80, can be considered as the length of the hypotenuse. The sum of the squares of the other two sides would be 49^2 + 64^2 = 2401 + 4096 = 6497.
However, 80^2 = 6400, which is less than 6497. Therefore, this set of numbers does not form a right triangle.
2. Set {45, 60, 75}:
Using the same approach, we can calculate the sum of the squares of the two smaller sides: 45^2 + 60^2 = 2025 + 3600 = 5625.
The square of the hypotenuse, 75^2, is also equal to 5625. Thus, this set of numbers does form a right triangle.
3. Set {48, 55, 72}:
Testing these numbers, we find that the sum of the squares of the smaller sides is 48^2 + 55^2 = 2304 + 3025 = 5329.
However, 72^2 = 5184, which is less than 5329. Therefore, this set of numbers does not form a right triangle.
4. Set {34, 56, 65}:
For these numbers, the sum of the squares of the two smaller sides is 34^2 + 56^2 = 1156 + 3136 = 4292.
Checking the square of the largest side, 65^2, we find that it is equal to 4225.
Since 4292 is not equal to 4225, this set of numbers does not form a right triangle.
In conclusion, only the set of numbers {45, 60, 75} can represent the sides of a right triangle.
this isnt working
You can save yourself some calculation by learning a few of the basic Pythagorean triples, such as
3-4-5, 5-12-13, 8-15-17, 7-24-25
and their multiples.