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Drag each triangle to the correct category.

For each set of side lengths, classify the triangle based on whether it is a right triangle.

Side lengths:
[tex]$
4, 4, \sqrt{32} \quad 4, 4, \sqrt{24} \quad 7, 8, 15 \quad 4, 10, 11 \quad 8, 7, \sqrt{113}
$[/tex]

\begin{tabular}{|l|l|}
\hline
Right Triangle & Not a Right Triangle \\
\hline
& \\
& \\
& \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
To classify each triangle based on whether it is a right triangle, we need to check if it satisfies the conditions of the Pythagorean theorem. A triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(c\)[/tex] is the hypotenuse, the longest side) is a right triangle if [tex]\(a^2 + b^2 = c^2\)[/tex].

Here are the given triangles with their side lengths:

1. [tex]\((4, 4, \sqrt{32})\)[/tex]
2. [tex]\((4, 4, \sqrt{24})\)[/tex]
3. [tex]\((7, 8, 15)\)[/tex]
4. [tex]\((4, 10, 11)\)[/tex]
5. [tex]\((8, 7, \sqrt{113})\)[/tex]

Let's classify these triangles step-by-step:

Right Triangle:

1. Triangle with sides (4, 4, √32):
- The side lengths are 4, 4, and √32.
- Calculate [tex]\(4^2 + 4^2 = 16 + 16 = 32\)[/tex].
- [tex]\( (\sqrt{32})^2 = 32\)[/tex].
- Since [tex]\(4^2 + 4^2 = \sqrt{32}^2\)[/tex], this is a right triangle.

2. Triangle with sides (8, 7, √113):
- The side lengths are 8, 7, and √113.
- Calculate [tex]\(8^2 + 7^2 = 64 + 49 = 113\)[/tex].
- [tex]\( (\sqrt{113})^2 = 113\)[/tex].
- Since [tex]\(8^2 + 7^2 = \sqrt{113}^2\)[/tex], this is a right triangle.

Not a Right Triangle:

1. Triangle with sides (4, 4, √24):
- The side lengths are 4, 4, and √24.
- Calculate [tex]\(4^2 + 4^2 = 16 + 16 = 32\)[/tex].
- [tex]\( (\sqrt{24})^2 = 24\)[/tex].
- Since [tex]\(4^2 + 4^2 \neq \sqrt{24}^2\)[/tex], this is not a right triangle.

2. Triangle with sides (7, 8, 15):
- The side lengths are 7, 8, and 15.
- Calculate [tex]\(7^2 + 8^2 = 49 + 64 = 113\)[/tex].
- Calculate [tex]\(15^2 = 225\)[/tex].
- Since [tex]\(7^2 + 8^2 \neq 15^2\)[/tex], this is not a right triangle.

3. Triangle with sides (4, 10, 11):
- The side lengths are 4, 10, and 11.
- Calculate [tex]\(4^2 + 10^2 = 16 + 100 = 116\)[/tex].
- Calculate [tex]\(11^2 = 121\)[/tex].
- Since [tex]\(4^2 + 10^2 \neq 11^2\)[/tex], this is not a right triangle.

So, our categorized triangles are:

[tex]\[ \begin{array}{|l|l|} \hline \text{Right Triangle} & \text{Not a Right Triangle} \\ \hline (4, 4, \sqrt{32}) & (4, 4, \sqrt{24}) \\ (8, 7, \sqrt{113}) & (7, 8, 15) \\ & (4, 10, 11) \\ \hline \end{array} \][/tex]
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