Question

here are the endpoints of the segments \\( \overline { p q } \\), \\( \overline { w x } \\), and \\( \overline { y z } \\).
\\( p ( - 6,7 ), q ( - 4,4 ) \\)
\\( w ( - 2, - 4 ), x ( 1, - 2 ) \\)
\\( y ( 4,2 ), z ( 5, - 2 ) \\)
follow the directions below.
(a) find the length of each segment.
give an exact answer (not a decimal approximation).
\\( p q = \\)
\\( w x = \\)
\\( y z = \\)
(b) check all statements that are true below.
\\( \square \overline { p q } \cong \overline { w x } \\)
\\( \square \overline { p q } \cong \overline { y z } \\)
\\( \square \overline { w x } \cong \overline { y z } \\)
\\( \square \\) none of these are true

Explanation:

Step1: Calculate the length of \(PQ\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(P(-6,7)\) and \(Q(-4,4)\), \(x_1=-6,y_1 = 7,x_2=-4,y_2 = 4\).
\(PQ=\sqrt{(-4+6)^2+(4 - 7)^2}=\sqrt{(2)^2+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}\)

Step2: Calculate the length of \(WX\)

For \(W(-2,-4)\) and \(X(1,-2)\), \(x_1=-2,y_1=-4,x_2 = 1,y_2=-2\).
\(WX=\sqrt{(1 + 2)^2+(-2+4)^2}=\sqrt{(3)^2+(2)^2}=\sqrt{9+4}=\sqrt{13}\)

Step3: Calculate the length of \(YZ\)

For \(Y(4,2)\) and \(Z(5,-2)\), \(x_1 = 4,y_1=2,x_2=5,y_2=-2\).
\(YZ=\sqrt{(5 - 4)^2+(-2 - 2)^2}=\sqrt{(1)^2+(-4)^2}=\sqrt{1 + 16}=\sqrt{17}\)

Step4: Check the congruence of segments

Since \(PQ=\sqrt{13}\), \(WX=\sqrt{13}\), and \(YZ=\sqrt{17}\).
Two segments are congruent if their lengths are equal. So \(\overline{PQ}\cong\overline{WX}\)

Answer:

(a) \(PQ=\sqrt{13}\), \(WX=\sqrt{13}\), \(YZ=\sqrt{17}\)
(b) \(\overline{PQ}\cong\overline{WX}\)