Respuesta :
Given:
A(2, -1), B(5, -1), C(5, 3)
Let's determine the if the triangle ABC is a right triangle.
To determine if it is a right triangle, let's find the length of each side by using distance formula, then apply Pythagorean Theorem.
Apply the distance formula:
[tex]d=\sqrt[]{(x2-x1)^2+(y2-y1)^2}[/tex]Thus, we have:
Length of AB:
A(2, -1), B(5, -1)
[tex]\begin{gathered} AB=\sqrt[]{(5-2)^2+(-1-(-1))^2} \\ \\ AB=\sqrt[]{(3)^2+(0)^2}=\sqrt[]{3^2}=3 \end{gathered}[/tex]Length of BC:
B(5, -1), C(5, 3)
[tex]\begin{gathered} BC=\sqrt[]{(5-5)^2+(3-(-1))^2} \\ \\ BC=\sqrt[]{(0)^2+(3+1)^2}=\sqrt[]{0+4^2}=4 \end{gathered}[/tex]Length of AC:
A(2, -1), C(5, 3)
[tex]\begin{gathered} AC=\sqrt[]{(5-2)^2+(3-(-1))^2} \\ \\ AC=\sqrt[]{(3)^2+(3+1)^2} \\ \\ AC=\sqrt[]{9+16}=\sqrt{25}=5 \end{gathered}[/tex]Thus, we have the following side lengths of triangle ABC:
AB = 3
BC = 4
AC = 5
APply Pythagorean Theorem:
[tex]c^2=a^2+b^2[/tex]Where:
a = 3
b = 4
c = 5
Let's verify if the equation is true:
[tex]\begin{gathered} c^2=a^2+b^2 \\ \\ 5^2=3^2+4^2 \\ \\ 25=9+16 \\ \\ 25=25 \end{gathered}[/tex]SInce the left hand side equals the right hand side, the equation is true.
Therefore, the triangle ABC is a right triangle.
ANSWER:
Yes, the triangle ABC is a right triangle.
