Let's find the solution step by step.
### Step 1: Identify the Coordinates of the Vertices
We have two triangles to consider:
- For triangle [tex]\(\triangle ABC\)[/tex], the vertices are:
- [tex]\(A(1, 7)\)[/tex]
- [tex]\(B(-2, 2)\)[/tex]
- [tex]\(C(4, 2)\)[/tex]
- For triangle [tex]\(\triangle ABD\)[/tex], the vertices are:
- [tex]\(A(1, 7)\)[/tex]
- [tex]\(B(-2, 2)\)[/tex]
- [tex]\(D(1, 2)\)[/tex]
### Step 2: Calculate the Lengths of the Sides of [tex]\(\triangle ABC\)[/tex]
To identify the longest side in [tex]\(\triangle ABC\)[/tex], we need to calculate the distances between each pair of vertices:
1. Length of [tex]\(AB\)[/tex]:
[tex]\( \sqrt{(1 - (-2))^2 + (7 - 2)^2} \)[/tex]
[tex]\( = \sqrt{(1 + 2)^2 + (7 - 2)^2} \)[/tex]
[tex]\( = \sqrt{3^2 + 5^2} \)[/tex]
[tex]\( = \sqrt{9 + 25} \)[/tex]
[tex]\( = \sqrt{34} \)[/tex]
[tex]\( \approx 5.83 \)[/tex] units
2. Length of [tex]\(BC\)[/tex]:
[tex]\( \sqrt{(4 - (-2))^2 + (2 - 2)^2} \)[/tex]
[tex]\( = \sqrt{(4 + 2)^2 + 0^2} \)[/tex]
[tex]\( = \sqrt{6^2 + 0} \)[/tex]
[tex]\( = \sqrt{36} \)[/tex]
[tex]\( = 6 \)[/tex] units
3. Length of [tex]\(AC\)[/tex]:
[tex]\( \sqrt{(4 - 1)^2 + (2 - 7)^2} \)[/tex]
[tex]\( = \sqrt{(4 - 1)^2 + (2 - 7)^2} \)[/tex]
[tex]\( = \sqrt{3^2 + (-5)^2} \)[/tex]
[tex]\( = \sqrt{9 + 25} \)[/tex]
[tex]\( = \sqrt{34} \)[/tex]
[tex]\( \approx 5.83 \)[/tex] units
Comparing the lengths, the longest side in [tex]\(\triangle ABC\)[/tex] is:
[tex]\[ \boxed{6} \][/tex] units.
### Step 3: Calculate the Lengths of the Sides of [tex]\(\triangle ABD\)[/tex]
To identify the longest side in [tex]\(\triangle ABD\)[/tex], we need to calculate the distances between each pair of vertices:
1. Length of [tex]\(AD\)[/tex]:
[tex]\( \sqrt{(1 - 1)^2 + (7 - 2)^2} \)[/tex]
[tex]\( = \sqrt{0 + 5^2} \)[/tex]
[tex]\( = \sqrt{25} \)[/tex]
[tex]\( = 5 \)[/tex] units
2. Length of [tex]\(BD\)[/tex]:
[tex]\( \sqrt{(-2 - 1)^2 + (2 - 2)^2} \)[/tex]
[tex]\( = \sqrt{(-2 - 1)^2 + 0} \)[/tex]
[tex]\( = \sqrt{(-3)^2} \)[/tex]
[tex]\( = \sqrt{9} \)[/tex]
[tex]\( = 3 \)[/tex] units
Comparing the lengths we have already computed, the longest side in [tex]\(\triangle ABD\)[/tex] is:
[tex]\[ \boxed{5.83} \][/tex] units.
Therefore, the correct answers to fill in the blanks are:
1. The measure of the longest side of [tex]\(\triangle ABC\)[/tex] is [tex]\( \boxed{6} \)[/tex] units.
2. The measure of the longest side of [tex]\(\triangle ABD\)[/tex] is [tex]\( \boxed{5.83} \)[/tex] units.
