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Quick Check

Multiple Choice: A student measured the sides of the triangle below.
[tex]\[ a = 3, \quad b = 4, \quad c = 5 \][/tex]

Using the Pythagorean Theorem:
[tex]\[
\begin{aligned}
a^2 + b^2 & = c^2 \\
3^2 + 4^2 & = 5^2 \\
9 + 16 & = 25 \\
25 & = 25
\end{aligned}
\][/tex]

Therefore, the triangle is a right triangle.

The conclusion uses which theorem?

A. Pythagorean Theorem
B. Converse of Pythagorean Theorem


Sagot :

To determine which theorem is used to conclude that the given triangle is a right triangle, let's carefully examine the steps and the reasoning behind them.

We start with the given sides of the triangle:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 5 \)[/tex]

We then apply the Pythagorean Theorem, which states that in a right triangle with legs [tex]\( a \)[/tex] and [tex]\( b \)[/tex], and hypotenuse [tex]\( c \)[/tex], the following equation holds true:
[tex]\[ a^2 + b^2 = c^2 \][/tex]

Substituting the given side lengths into the equation, we get:
[tex]\[ 3^2 + 4^2 = 5^2 \][/tex]

Next, we perform the calculations:
[tex]\[ 9 + 16 = 25 \][/tex]

Since both sides of the equation are equal:
[tex]\[ 25 = 25 \][/tex]

This equality confirms that the triangle with sides 3, 4, and 5 satisfies the condition of the Pythagorean Theorem. In essence, what we are concluding here is that these measurements form a right triangle.

To explicitly state the conclusion drawn from this calculation, we recognize that verifying [tex]\( a^2 + b^2 = c^2 \)[/tex] and thereby determining that the triangle is right-angled uses the Pythagorean Theorem.

So, the correct conclusion uses:

a) Pythagorean Theorem