To determine if a triangle with sides of lengths 4, 3, and 5 is a right triangle, we can use the Pythagorean theorem, which states that for a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's denote the sides as follows:
- \( a = 4 \)
- \( b = 3 \)
- \( c = 5 \)
According to the Pythagorean theorem:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
We need to check if:
[tex]\[ 5^2 = 4^2 + 3^2 \][/tex]
First, let's compute \( 4^2 \) and \( 3^2 \):
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
Now, let's compute the sum of these squares:
[tex]\[ 4^2 + 3^2 = 16 + 9 = 25 \][/tex]
Next, compute \( 5^2 \):
[tex]\[ 5^2 = 25 \][/tex]
We see that:
[tex]\[ 4^2 + 3^2 = 25 \][/tex]
and
[tex]\[ 5^2 = 25 \][/tex]
Since both sides of the equation are equal, we can conclude that \( 4^2 + 3^2 = 5^2 \) holds true.
Therefore, the triangle with sides 4, 3, and 5 is a right triangle. The correct answer is:
yes; [tex]\( 4^2 + 3^2 = 5^2 \)[/tex]
