Question
The isosceles triangle has a base that measures 14 units.
A triangle has a base length of 14. The other 2 sides have a length of y.
The value of y, the length of each leg, must be
equal to 7.
between 7 and 14.
greater than 7.
between 14 and 28.
A triangle has a base length of 14. The other 2 sides have a length of y.
The value of y, the length of each leg, must be
equal to 7.
between 7 and 14.
greater than 7.
between 14 and 28.
Answers
Answered by
GPT-4o mini
To determine the possible lengths for the legs (y) of an isosceles triangle with a base of 14 units, we need to apply the triangle inequality theorem.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For our isosceles triangle, with legs of length \(y\) and a base of length 14 units, we can set up the following inequalities:
1. \(y + y > 14\) (the sum of the two legs must be greater than the base)
2. \(y + 14 > y\) (the sum of one leg and the base must be greater than the other leg, which reduces to \(14 > 0\), always true)
3. \(y + 14 > y\) (same logic as above, always true)
From the first inequality:
\[
2y > 14
\]
Dividing by 2:
\[
y > 7
\]
So the value of \(y\) must be greater than 7.
Next, we also consider the maximum possible value for \(y\). Since \(y\) must also form a triangle with the base, the following condition must hold:
\[
y < 14 + y
\]
This is true for any positive value of \(y\) because it becomes a tautology. However, we also have to consider:
1. \(y + y > 14\) (this is satisfied when \(y > 7\)),
2. and also that \(y\) can potentially create a triangle with the base.
Considering these factors, while \(y\) can be any value greater than 7, if \(y\) were to equal 14, it would reduce the triangle to a straight line, so \(y\) must also be less than 14.
Thus, the final conclusion is:
The value of \(y\), the length of each leg, must be:
- Greater than 7 and less than 14.
In interval notation, \(y\) must be in the range:
\[
(7, 14)
\]
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For our isosceles triangle, with legs of length \(y\) and a base of length 14 units, we can set up the following inequalities:
1. \(y + y > 14\) (the sum of the two legs must be greater than the base)
2. \(y + 14 > y\) (the sum of one leg and the base must be greater than the other leg, which reduces to \(14 > 0\), always true)
3. \(y + 14 > y\) (same logic as above, always true)
From the first inequality:
\[
2y > 14
\]
Dividing by 2:
\[
y > 7
\]
So the value of \(y\) must be greater than 7.
Next, we also consider the maximum possible value for \(y\). Since \(y\) must also form a triangle with the base, the following condition must hold:
\[
y < 14 + y
\]
This is true for any positive value of \(y\) because it becomes a tautology. However, we also have to consider:
1. \(y + y > 14\) (this is satisfied when \(y > 7\)),
2. and also that \(y\) can potentially create a triangle with the base.
Considering these factors, while \(y\) can be any value greater than 7, if \(y\) were to equal 14, it would reduce the triangle to a straight line, so \(y\) must also be less than 14.
Thus, the final conclusion is:
The value of \(y\), the length of each leg, must be:
- Greater than 7 and less than 14.
In interval notation, \(y\) must be in the range:
\[
(7, 14)
\]