Question

determine which of the triangles can be described as a right triangle. recall that a right triangle has one 90° interior angle.

Explanation:

Step1: Calculate the lengths of the sides of triangle \(ABC\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(AB\): \(x_1 = 0,y_1 = 12,x_2 = 20,y_2 = 21\)
\(AB=\sqrt{(20 - 0)^2+(21 - 12)^2}=\sqrt{400 + 81}=\sqrt{481}\)
For \(AC\): \(x_1 = 0,y_1 = 12,x_2 = 18,y_2 = 9\)
\(AC=\sqrt{(18 - 0)^2+(9 - 12)^2}=\sqrt{324+9}=\sqrt{333}\)
For \(BC\): \(x_1 = 20,y_1 = 21,x_2 = 18,y_2 = 9\)
\(BC=\sqrt{(18 - 20)^2+(9 - 21)^2}=\sqrt{4 + 144}=\sqrt{148}\)
Check \(AB^{2}+AC^{2}
eq BC^{2},AB^{2}+BC^{2}
eq AC^{2},AC^{2}+BC^{2}
eq AB^{2}\)

Step2: Calculate the lengths of the sides of triangle \(DEF\)

For \(DE\): \(x_1=-12,y_1 = 0,x_2=-32,y_2=-9\)
\(DE=\sqrt{(-32 + 12)^2+(-9 - 0)^2}=\sqrt{400+81}=\sqrt{481}\)
For \(DF\): \(x_1=-12,y_1 = 0,x_2 = 0,y_2=-27\)
\(DF=\sqrt{(0 + 12)^2+(-27 - 0)^2}=\sqrt{144 + 729}=\sqrt{873}\)
For \(EF\): \(x_1=-32,y_1=-9,x_2 = 0,y_2=-27\)
\(EF=\sqrt{(0 + 32)^2+(-27 + 9)^2}=\sqrt{1024+324}=\sqrt{1348}\)
Check \(DE^{2}+DF^{2}
eq EF^{2},DE^{2}+EF^{2}
eq DF^{2},DF^{2}+EF^{2}
eq DE^{2}\)

Step3: Calculate the lengths of the sides of triangle \(JKL\)

For \(JK\): \(x_1 = 10,y_1=-10,x_2 = 15,y_2=-5\)
\(JK=\sqrt{(15 - 10)^2+(-5 + 10)^2}=\sqrt{25 + 25}=\sqrt{50}\)
For \(JL\): \(x_1 = 10,y_1=-10,x_2 = 30,y_2=-20\)
\(JL=\sqrt{(30 - 10)^2+(-20 + 10)^2}=\sqrt{400+100}=\sqrt{500}\)
For \(KL\): \(x_1 = 15,y_1=-5,x_2 = 30,y_2=-20\)
\(KL=\sqrt{(30 - 15)^2+(-20 + 5)^2}=\sqrt{225+225}=\sqrt{450}\)
Check \(JK^{2}+KL^{2}=50 + 450=500=JL^{2}\)

Answer:

Triangle \(JKL\) is a right - triangle.