1. Determine which set of three numbers could represent the sides of a right triangle.
a. 9, 41, 41
b. 9, 40, 41
c. 8, 41, 40
d. 10, 40, 42
is the answer b?
2. Determine which set of three numbers could not represent the sides of a right triangle.
a. 3, 4, 5
b. 21, 28, 35
c. 14, 19, 24
d. 30, 40, 50
is the answer d?
1. Correct
2. If a is representative, so is d. c is correct.
To determine if a set of three numbers can represent the sides of a right triangle, you can use the Pythagorean theorem. According to the theorem, in a right triangle, the sum of the squares of the two smaller sides should be equal to the square of the longest side (hypotenuse).
1. Let's check each set of numbers using the Pythagorean theorem:
a. 9, 41, 41
Using the theorem: 9^2 + 41^2 = 1682^2 ≠ 41^2
Therefore, a is not a set of sides that can represent a right triangle.
b. 9, 40, 41
Using the theorem: 9^2 + 40^2 = 1681^2 = 41^2
Therefore, b is a set of sides that can represent a right triangle.
c. 8, 41, 40
Using the theorem: 8^2 + 41^2 = 1681^2 ≠ 40^2
Therefore, c is not a set of sides that can represent a right triangle.
d. 10, 40, 42
Using the theorem: 10^2 + 40^2 = 1700^2 ≠ 42^2
Therefore, d is not a set of sides that can represent a right triangle.
So, the correct answer is b.
2. Let's check each set of numbers again:
a. 3, 4, 5
Using the theorem: 3^2 + 4^2 = 5^2
Therefore, a is a set of sides that can represent a right triangle.
b. 21, 28, 35
Using the theorem: 21^2 + 28^2 = 35^2
Therefore, b is a set of sides that can represent a right triangle.
c. 14, 19, 24
Using the theorem: 14^2 + 19^2 = 24^2 ≠ 19^2
Therefore, c is not a set of sides that can represent a right triangle.
d. 30, 40, 50
Using the theorem: 30^2 + 40^2 = 50^2
Therefore, d is a set of sides that can represent a right triangle.
So, the correct answer is c, not d.
1. To determine which set of three numbers could represent the sides of a right triangle, we can use the Pythagorean theorem. According to the theorem, in a right triangle, the sum of the squares of the two shorter sides (a and b) is equal to the square of the longest side (c).
Let's try each set of numbers:
a. 9, 41, 41
The sum of the squares of the two shorter sides: 9^2 + 41^2 = 1682
The square of the longest side: 41^2 = 1681
These values are not equal, so this set does not represent the sides of a right triangle.
b. 9, 40, 41
The sum of the squares of the two shorter sides: 9^2 + 40^2 = 1681
The square of the longest side: 41^2 = 1681
These values are equal, so this set represents the sides of a right triangle.
c. 8, 41, 40
The sum of the squares of the two shorter sides: 8^2 + 41^2 = 1745
The square of the longest side: 40^2 = 1600
These values are not equal, so this set does not represent the sides of a right triangle.
d. 10, 40, 42
The sum of the squares of the two shorter sides: 10^2 + 40^2 = 1700
The square of the longest side: 42^2 = 1764
These values are not equal, so this set does not represent the sides of a right triangle.
Therefore, the answer is b. Set b (9, 40, 41) could represent the sides of a right triangle.
2. Similar to the first question, we will use the Pythagorean theorem to determine the set of three numbers that could not represent the sides of a right triangle.
Let's try each set of numbers:
a. 3, 4, 5
The sum of the squares of the two shorter sides: 3^2 + 4^2 = 25
The square of the longest side: 5^2 = 25
These values are equal, so this set represents the sides of a right triangle.
b. 21, 28, 35
The sum of the squares of the two shorter sides: 21^2 + 28^2 = 1225 + 784 = 2009
The square of the longest side: 35^2 = 1225
These values are not equal, so this set does not represent the sides of a right triangle.
c. 14, 19, 24
The sum of the squares of the two shorter sides: 14^2 + 19^2 = 196 + 361 = 557
The square of the longest side: 24^2 = 576
These values are not equal, so this set does not represent the sides of a right triangle.
d. 30, 40, 50
The sum of the squares of the two shorter sides: 30^2 + 40^2 = 900 + 1600 = 2500
The square of the longest side: 50^2 = 2500
These values are equal, so this set represents the sides of a right triangle.
Therefore, the answer is b. Set b (21, 28, 35) could not represent the sides of a right triangle.