If two sides of a triangle are 3 cm and 5 cm in length, which could NOT be the measure of the third side?
Responses
A 4 cm4 cm
B 2 cm2 cm
C 5 cm5 cm
D 3 cm
why
It is quite possible to have an isosceles triangle with two sides of length 5, and the base of length 3.
All you have to do is check the longest side to be sure it is between the difference and the sum of the other two sides. In this case, choice B cannot be used, since if two sides are 2 and 3, they will just collapse if laid against the side of length 5. That is,
3-2 < 5 < 3+2
is not true.
I apologize, you are correct. The triangle inequality theorem can be used to eliminate option C, but option B is actually a possible measure for the third side of an isosceles triangle with sides of length 3 cm and 5 cm. Thank you for bringing this to my attention.
To solve this problem, we can use the triangle inequality theorem, which states that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
In this case, we have two sides with lengths of 3 cm and 5 cm. Let's check each answer choice to see if it satisfies the triangle inequality theorem:
A. 4 cm: To satisfy the triangle inequality theorem, the sum of the lengths of the two known sides (3 cm + 5 cm) must be greater than the length of the third side. 3 cm + 5 cm = 8 cm, which is greater than 4 cm. Therefore, 4 cm could be the measure of the third side.
B. 2 cm: Using the same logic, we find that 3 cm + 5 cm = 8 cm, which is greater than 2 cm. Therefore, 2 cm could be the measure of the third side.
C. 5 cm: Now, let's consider the given answer choice. If the third side were also 5 cm, the sum of the two known sides (3 cm + 5 cm) would be equal to the length of the third side. This means that the triangle would be a straight line, which is not possible. Therefore, 5 cm could NOT be the measure of the third side.
D. 3 cm: Similarly, if the third side were also 3 cm, the sum of the two known sides (3 cm + 5 cm) would be equal to the length of the third side. Again, this would result in a straight line, which is not a valid triangle. Therefore, 3 cm could NOT be the measure of the third side.
In conclusion, the length of the third side of the triangle could not be 5 cm (Option C) or 3 cm (Option D).
C 5 cm5 cm
This is because of the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Applying this to the given triangle, the sum of the lengths of the two shorter sides (3 cm and 5 cm) is 8 cm, which means the length of the third side must be less than 8 cm to satisfy the triangle inequality.
Option C, 5 cm, is equal to the length of one of the given sides and therefore does not satisfy the triangle inequality.