To find the measurement of the third side of the triangle, let's break down and organize the given information step by step.
First, we need to express the given lengths properly:
- The total sum of the sides of the triangle is [tex]\(18 \frac{2}{9} \; cm\)[/tex].
Converting this mixed number to an improper fraction, we get:
[tex]\[
18 \frac{2}{9} = 18 + \frac{2}{9} = \frac{18 \times 9}{9} + \frac{2}{9} = \frac{162}{9} + \frac{2}{9} = \frac{164}{9}
\][/tex]
- One of the sides of the triangle is [tex]\(8 \frac{2}{3} \; cm\)[/tex].
Converting this mixed number to an improper fraction, we get:
[tex]\[
8 \frac{2}{3} = 8 + \frac{2}{3} = \frac{8 \times 3}{3} + \frac{2}{3} = \frac{24}{3} + \frac{2}{3} = \frac{26}{3}
\][/tex]
- Another side is [tex]\(4 \frac{1}{6} \; cm\)[/tex].
Converting this mixed number to an improper fraction, we get:
[tex]\[
4 \frac{1}{6} = 4 + \frac{1}{6} = \frac{4 \times 6}{6} + \frac{1}{6} = \frac{24}{6} + \frac{1}{6} = \frac{25}{6}
\][/tex]
Next, we need to find the value of the third side. The formula used to find the third side ([tex]\( \text{Third side} \)[/tex]) of the triangle is:
[tex]\[
\text{Third side} = \text{Total sum of sides} - \text{Side 1} - \text{Side 2}
\][/tex]
Let's plug in the values we have:
[tex]\[
\text{Total sum of sides} = \frac{164}{9}
\][/tex]
[tex]\[
\text{Side 1} = \frac{26}{3} = \frac{26 \times 2}{3 \times 2} = \frac{52}{6}
\][/tex]
[tex]\[
\text{Side 2} = \frac{25}{6}
\][/tex]
Since Side 1 and Side 2 need a common denominator for subtraction, we convert both to have 6 as the denominator:
Now, summing Side 1 and Side 2:
[tex]\[
\text{Side 1} + \text{Side 2} = \frac{52}{6} + \frac{25}{6} = \frac{52+25}{6} = \frac{77}{6}
\][/tex]
Next, convert the total sum of the sides to have a common denominator of 6:
[tex]\[
\frac{164}{9} = \frac{164 \times 2}{9 \times 2} = \frac{328}{18} = \frac{328}{3 \times 6} = \frac{328}{18} = \frac{164 \times 2}{9 \times 2} = \frac{328}{18} \left(\text{Converting to have a denominator of 18}\right)
\][/tex]
Lastly, we find [tex]\(\frac{164}{9}\)[/tex] expressed with a common denominator of 18 to have:
[tex]\[
\text{Third side} = \frac{164}{9} - \frac{77}{6}
\][/tex]
[tex]\[
\frac{328}{18} - \frac{231}{18} = \frac{328-231}{18} = \frac{97}{18}\\
Finally, \boxed(Third side) will be 5.389.
This results in the third side being:
\[ \text{Third side} = 5.389 \; cm \][/tex]
Hence, the measurement of the third side is [tex]\(5.389 \; cm\)[/tex].
