Show that the points form the vertices of the indicated polygon. Right triangle: $( 5,2 ) , ( 2,4 ) , ( - 2 , - 2 )$
A) $( \sqrt { 11 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
B) $( \sqrt { 13 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
C) $( \sqrt { 29 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
D) $( \sqrt { 20 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
E) $( \sqrt { 7 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$

Question

Show that the points form the vertices of the indicated polygon. Right triangle: $( 5,2 ) , ( 2,4 ) , ( - 2 , - 2 )$ 
A) $( \sqrt { 11 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
B) $( \sqrt { 13 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
C) $( \sqrt { 29 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
D) $( \sqrt { 20 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$
E) $( \sqrt { 7 } ) ^ { 2 } + ( \sqrt { 52 } ) ^ { 2 } = ( \sqrt { 65 } ) ^ { 2 }$

Quizplus · Accepted Answer

To show that the points form the vertices of a right triangle, we calculate the distances between the points and verify if they satisfy the Pythagorean theorem. The distances between the points $(5,2)$, $(2,4)$, and $(-2,-2)$ are calculated as follows:1. Distance between $(5,2)$ and $(2,4)$: $\sqrt{(5-2)^2 + (2-4)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}$.2. Distance between $(5,2)$ and $(-2,-2)$: $\sqrt{(5-(-2))^2 + (2-(-2))^2} = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65}$.3. Distance between $(2,4)$ and $(-2,-2)$: $\sqrt{(2-(-2))^2 + (4-(-2))^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}$.Checking the Pythagorean theorem with these distances, we find that $(\sqrt{13})^2 + (\sqrt{52})^2 = 13 + 52 = 65 = (\sqrt{65})^2$, which matches option B. This confirms that the points form a right triangle.

Show That the Points Form the Vertices of the Indicated (5,2),(2,4),(−2,−2)( 5,2 ) , ( 2,4 ) , ( - 2 , - 2 )(5,2),(2,4),(−2,−2)

Show That the Points Form the Vertices of the Indicated $( 5,2 ) , ( 2,4 ) , ( - 2 , - 2 )$