Which of the following cannot be the lengths of a 30°-60°-90° triangle?
A: 10/3, 20/3, 10/3 √3
B: 3, 3/2, 3√3
C: 11, 22, 11√3
D: 5/2, 5, 5/2 √3
I believe the answer is B, but I'm not sure.
To determine which of the given options cannot be the lengths of a 30°-60°-90° triangle, we need to use the property of this type of triangle.
In a 30°-60°-90° triangle, the ratio of the lengths of the sides is:
1 : √3 : 2
Let's evaluate each option to see if it satisfies this ratio:
A: 10/3, 20/3, 10/3 √3
Dividing each side length by the corresponding ratio gives:
(10/3)/(1) = 10/3
(20/3)/(√3) = (20/3) / (√3) * (√3) / (√3) = (20/3)√3 / 3 = 20√3 / 9
(10/3 √3)/(2) = (10/3 √3)/(2) * (√3)/(√3) = (10/3) / (2√3) = 10/(6√3) = 5√3 / 9
So, the lengths in option A satisfy the ratio and could be the lengths of a 30°-60°-90° triangle.
B: 3, 3/2, 3√3
Using the same process, we get:
(3)/(1) = 3
(3/2)/(√3) = (3/2) / (√3) * (√3) / (√3) = (3/2)√3 / 3 = 3√3 / 6 = √3 / 2
(3√3)/(2) = (3√3)/(2)
So, the lengths in option B also satisfy the ratio and could be the lengths of a 30°-60°-90° triangle.
C: 11, 22, 11√3
Evaluating the lengths:
(11)/(1) = 11
(22)/(√3) = (22) / (√3) * (√3) / (√3) = (22)√3 / 3
(11√3)/(2) = (11√3)/(2)
The lengths in option C also satisfy the ratio and could be the lengths of a 30°-60°-90° triangle.
D: 5/2, 5, 5/2 √3
Evaluating the lengths:
(5/2)/(1) = 5/2
(5)/(√3) = (5) / (√3) * (√3) / (√3) = (5)√3 / 3
(5/2 √3)/(2) = (5/2 √3)/(2) * (√3)/(√3) = (5/2) / (2√3) = 5/(4√3) = 5√3 / 12
So, the lengths in option D also satisfy the ratio and could be the lengths of a 30°-60°-90° triangle.
Based on our calculations, none of the given options would be incorrect lengths for a 30°-60°-90° triangle. Therefore, your original suspicion that option B is incorrect is not correct. All the given options could be the lengths of a 30°-60°-90° triangle.
To determine which of the given options cannot be the lengths of a 30°-60°-90° triangle, we need to understand the properties of this specific type of triangle.
In a 30°-60°-90° triangle, the ratio of the lengths of the sides is always 1 : √3 : 2.
Option A: 10/3, 20/3, 10/3 √3
To check if these lengths could form a 30°-60°-90° triangle, we divide each length by the smallest side (10/3) to get the ratios:
(10/3) / (10/3) = 1
(20/3) / (10/3) = 2
(10/3 √3) / (10/3) = √3
As these ratios match the typical 30°-60°-90° triangle ratios, Option A could be the lengths of a 30°-60°-90° triangle.
Option B: 3, 3/2, 3√3
Following the same process, we divide each length by the smallest side (3/2):
3 / (3/2) = 2
(3/2) / (3/2) = 1
(3√3) / (3/2) = 2√3
Here, the ratios don't match the typical 30°-60°-90° triangle ratios, which means Option B cannot be the lengths of a 30°-60°-90° triangle.
Option C: 11, 22, 11√3
Again, dividing each length by the smallest side (11):
11 / 11 = 1
(22) / 11 = 2
(11√3) / 11 = √3
These ratios match the typical 30°-60°-90° triangle ratios, so Option C could be the lengths of a 30°-60°-90° triangle.
Option D: 5/2, 5, 5/2√3
Dividing each length by the smallest side (5/2):
(5/2) / (5/2) = 1
5 / (5/2) = 2
(5/2√3) / (5/2) = √3
The ratios match the typical 30°-60°-90° triangle ratios, so Option D could be the lengths of a 30°-60°-90° triangle.
Therefore, the option that cannot be the lengths of a 30°-60°-90° triangle is indeed Option B.