Question
Two sides of a triangle measure 4 and 6. Which inequality shows all the possible lengths of the third side, x?
A. 2/3<x<24
B. 4<x<6
C. 2<x<10
D. 3<x<12
A. 2/3<x<24
B. 4<x<6
C. 2<x<10
D. 3<x<12
Answers
GPT-4o mini
In a triangle, the lengths of the sides must satisfy the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides must be greater than the length of the third side.
Let the lengths of the two sides of the triangle be \(a = 4\) and \(b = 6\), and let the length of the third side be \(x\). We need to establish the inequalities based on the triangle inequality theorem:
1. \(a + b > x\)
\[
4 + 6 > x \implies 10 > x \implies x < 10
\]
2. \(a + x > b\)
\[
4 + x > 6 \implies x > 6 - 4 \implies x > 2
\]
3. \(b + x > a\)
\[
6 + x > 4 \implies x > 4 - 6 \implies x > -2 \quad (\text{This is always true for } x > 2)
\]
Combining these inequalities, we find:
\[
2 < x < 10
\]
Thus, the answer is:
**C. \(2 < x < 10\)**
Let the lengths of the two sides of the triangle be \(a = 4\) and \(b = 6\), and let the length of the third side be \(x\). We need to establish the inequalities based on the triangle inequality theorem:
1. \(a + b > x\)
\[
4 + 6 > x \implies 10 > x \implies x < 10
\]
2. \(a + x > b\)
\[
4 + x > 6 \implies x > 6 - 4 \implies x > 2
\]
3. \(b + x > a\)
\[
6 + x > 4 \implies x > 4 - 6 \implies x > -2 \quad (\text{This is always true for } x > 2)
\]
Combining these inequalities, we find:
\[
2 < x < 10
\]
Thus, the answer is:
**C. \(2 < x < 10\)**