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
