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Holly and Tamar completed the work in the table to determine if a triangle with side lengths of 40, 42, and 58 is a right triangle.

[tex]\[
\begin{tabular}{|c|c|}
\hline
\textbf{Holly's Work} & \textbf{Tamar's Work} \\
\hline
& \\
$(40 + 42)^2 = 58^2$ & $42^2 + 40^2 = 58^2$ \\
$82^2 = 3,364$ & $1,764 + 1,600 \stackrel{?}{=} 3,364$ \\
$6,724 \neq 3,364$ & $3,364 = 3,364$ \\
The triangle is not a right triangle. & The triangle is a right triangle. \\
\hline
\end{tabular}
\][/tex]

Which best describes the accuracy of their solutions?

Sagot :

GinnyAnswer
Let's analyze the methods and results step-by-step to determine the accuracy of Holly's and Tamar's solutions.

### Tamar's Work
Tamar uses the Pythagorean theorem to verify if the triangle is a right triangle. According to the Pythagorean theorem, for a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and hypotenuse [tex]\( c \)[/tex], the triangle is right if and only if:
[tex]\[ a^2 + b^2 = c^2 \][/tex]

For the given triangle:
- [tex]\( a = 40 \)[/tex]
- [tex]\( b = 42 \)[/tex]
- [tex]\( c = 58 \)[/tex]

1. Calculate [tex]\( a^2 \)[/tex]:
[tex]\[ 40^2 = 1600 \][/tex]

2. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ 42^2 = 1764 \][/tex]

3. Calculate [tex]\( c^2 \)[/tex]:
[tex]\[ 58^2 = 3364 \][/tex]

4. Check if [tex]\( a^2 + b^2 = c^2 \)[/tex]:
[tex]\[ 1600 + 1764 = 3364 \][/tex]

Since the left-hand side is equal to the right-hand side:
[tex]\[ 3364 = 3364 \][/tex]
Tamar concludes that the triangle is a right triangle.

### Holly's Work
Holly incorrectly applies a different method:
[tex]\[ (a + b)^2 = c^2 \][/tex]

For the given triangle:
- [tex]\( a = 40 \)[/tex]
- [tex]\( b = 42 \)[/tex]
- [tex]\( c = 58 \)[/tex]

1. Calculate [tex]\( a + b \)[/tex]:
[tex]\[ 40 + 42 = 82 \][/tex]

2. Calculate [tex]\( (a + b)^2 \)[/tex]:
[tex]\[ 82^2 = 6724 \][/tex]

3. Calculate [tex]\( c^2 \)[/tex]:
[tex]\[ 58^2 = 3364 \][/tex]

4. Check if [tex]\( (a + b)^2 = c^2 \)[/tex]:
[tex]\[ 6724 \neq 3364 \][/tex]

Since the left-hand side is not equal to the right-hand side:
[tex]\[ 6724 \neq 3364 \][/tex]
Holly concludes that the triangle is not a right triangle.

### Conclusion
- Tamar's Solution: Accurate. She correctly applies the Pythagorean theorem and verifies the side lengths correctly, concluding that the triangle is a right triangle.
- Holly's Solution: Inaccurate. She applies a faulty method unrelated to the properties of right triangles. Her conclusion that the triangle is not a right triangle is incorrect based on incorrect reasoning.

Given the correct numerical results and accurate application of the Pythagorean theorem, Tamar's method is reliable and correctly identifies the given triangle as a right triangle.
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