Answer :
To determine the correct identification of points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] on the number line using the values [tex]\(\sqrt{10}\)[/tex], [tex]\(\sqrt{13}\)[/tex], and [tex]\(\frac{22}{9}\)[/tex], we start by calculating and comparing these values.
The numerical values are:
1. [tex]\(\sqrt{10} \approx 3.1622776601683795\)[/tex]
2. [tex]\(\sqrt{13} \approx 3.605551275463989\)[/tex]
3. [tex]\(\frac{22}{9} \approx 2.4444444444444446\)[/tex]
Now, we will use these values to associate them with their respective points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] based on the provided choices:
1. Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{10}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{13}\)[/tex].
This choice claims:
- [tex]\(A \approx 2.4444444444444446\)[/tex]
- [tex]\(B \approx 3.1622776601683795\)[/tex]
- [tex]\(C \approx 3.605551275463989\)[/tex]
2. Point [tex]\(A\)[/tex] is [tex]\(\sqrt{10}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{13}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\frac{22}{9}\)[/tex].
This choice claims:
- [tex]\(A \approx 3.1622776601683795\)[/tex]
- [tex]\(B \approx 3.605551275463989\)[/tex]
- [tex]\(C \approx 2.4444444444444446\)[/tex]
3. Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{13}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{10}\)[/tex].
This choice claims:
- [tex]\(A \approx 2.4444444444444446\)[/tex]
- [tex]\(B \approx 3.605551275463989\)[/tex]
- [tex]\(C \approx 3.1622776601683795\)[/tex]
4. Point [tex]\(A\)[/tex] is [tex]\(\sqrt{10}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{13}\)[/tex].
This choice claims:
- [tex]\(A \approx 3.1622776601683795\)[/tex]
- [tex]\(B \approx 2.4444444444444446\)[/tex]
- [tex]\(C \approx 3.605551275463989\)[/tex]
Given the values for [tex]\(\sqrt{10}\)[/tex], [tex]\(\sqrt{13}\)[/tex], and [tex]\(\frac{22}{9}\)[/tex], we can now find the correct identification of points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] on the number line.
### Evaluation of Each Choice
- Comparing the values, [tex]\(\frac{22}{9} \approx 2.4444444444444446\)[/tex] is the smallest.
- [tex]\(\sqrt{10} \approx 3.1622776601683795\)[/tex] is next.
- [tex]\(\sqrt{13} \approx 3.605551275463989\)[/tex] is the largest.
From this information, we can see:
Choice (3) is correctly matches the values:
- Point [tex]\(A \approx 2.4444444444444446\)[/tex] matches [tex]\(\frac{22}{9}\)[/tex].
- Point [tex]\(B \approx 3.605551275463989\)[/tex] matches [tex]\(\sqrt{13}\)[/tex].
- Point [tex]\(C \approx 3.1622776601683795\)[/tex] matches [tex]\(\sqrt{10}\)[/tex].
Therefore, the correct identification of the points is:
Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{13}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{10}\)[/tex].
The numerical values are:
1. [tex]\(\sqrt{10} \approx 3.1622776601683795\)[/tex]
2. [tex]\(\sqrt{13} \approx 3.605551275463989\)[/tex]
3. [tex]\(\frac{22}{9} \approx 2.4444444444444446\)[/tex]
Now, we will use these values to associate them with their respective points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] based on the provided choices:
1. Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{10}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{13}\)[/tex].
This choice claims:
- [tex]\(A \approx 2.4444444444444446\)[/tex]
- [tex]\(B \approx 3.1622776601683795\)[/tex]
- [tex]\(C \approx 3.605551275463989\)[/tex]
2. Point [tex]\(A\)[/tex] is [tex]\(\sqrt{10}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{13}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\frac{22}{9}\)[/tex].
This choice claims:
- [tex]\(A \approx 3.1622776601683795\)[/tex]
- [tex]\(B \approx 3.605551275463989\)[/tex]
- [tex]\(C \approx 2.4444444444444446\)[/tex]
3. Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{13}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{10}\)[/tex].
This choice claims:
- [tex]\(A \approx 2.4444444444444446\)[/tex]
- [tex]\(B \approx 3.605551275463989\)[/tex]
- [tex]\(C \approx 3.1622776601683795\)[/tex]
4. Point [tex]\(A\)[/tex] is [tex]\(\sqrt{10}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{13}\)[/tex].
This choice claims:
- [tex]\(A \approx 3.1622776601683795\)[/tex]
- [tex]\(B \approx 2.4444444444444446\)[/tex]
- [tex]\(C \approx 3.605551275463989\)[/tex]
Given the values for [tex]\(\sqrt{10}\)[/tex], [tex]\(\sqrt{13}\)[/tex], and [tex]\(\frac{22}{9}\)[/tex], we can now find the correct identification of points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] on the number line.
### Evaluation of Each Choice
- Comparing the values, [tex]\(\frac{22}{9} \approx 2.4444444444444446\)[/tex] is the smallest.
- [tex]\(\sqrt{10} \approx 3.1622776601683795\)[/tex] is next.
- [tex]\(\sqrt{13} \approx 3.605551275463989\)[/tex] is the largest.
From this information, we can see:
Choice (3) is correctly matches the values:
- Point [tex]\(A \approx 2.4444444444444446\)[/tex] matches [tex]\(\frac{22}{9}\)[/tex].
- Point [tex]\(B \approx 3.605551275463989\)[/tex] matches [tex]\(\sqrt{13}\)[/tex].
- Point [tex]\(C \approx 3.1622776601683795\)[/tex] matches [tex]\(\sqrt{10}\)[/tex].
Therefore, the correct identification of the points is:
Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{13}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{10}\)[/tex].
