To determine the range of possible values for the third side of an acute triangle with sides measuring 10 cm and 16 cm, we need to consider both the triangle inequality conditions and the condition for the triangle being acute.
1. The Triangle Inequality Theorem:
- For any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the following must hold true:
[tex]\[
a + b > c
\][/tex]
[tex]\[
a + c > b
\][/tex]
[tex]\[
b + c > a
\][/tex]
Given sides [tex]\(a = 10\)[/tex] cm and [tex]\(b = 16\)[/tex] cm:
- Applying the inequality [tex]\( a + b > c \)[/tex]:
[tex]\[
10 + 16 > c \quad \Rightarrow \quad 26 > c \quad \Rightarrow \quad c < 26
\][/tex]
- Applying the inequality [tex]\( a + c > b \)[/tex]:
[tex]\[
10 + c > 16 \quad \Rightarrow \quad c > 6
\][/tex]
- Applying the inequality [tex]\( b + c > a \)[/tex]:
[tex]\[
16 + c > 10
\][/tex]
This inequality is always true for all [tex]\( c > 0 \)[/tex] given that [tex]\(c\)[/tex] must be greater than 6 (from the previous inequality).
So, combining the inequalities from above:
[tex]\[
6 < c < 26
\][/tex]
2. Condition for an Acute Triangle:
- For a triangle to be acute, the square of the length of each side must be less than the sum of the squares of the other two sides:
[tex]\[
c^2 < a^2 + b^2
\][/tex]
Given [tex]\(a = 10\)[/tex] cm and [tex]\(b = 16\)[/tex] cm, we calculate:
[tex]\[
c^2 < 10^2 + 16^2 \quad \Rightarrow \quad c^2 < 100 + 256 \quad \Rightarrow \quad c^2 < 356 \quad \Rightarrow \quad c < \sqrt{356} \approx 18.87
\][/tex]
Combining the conditions for the third side, we get:
[tex]\[
6 < c < 18.87
\][/tex]
Hence, the range of possible values for the third side of the triangle is:
Answer:
[tex]\[
6 < x < 18.9
\][/tex]
