Question

the segments \\(\overline{pq}\\), \\(\overline{wx}\\), and \\(\overline{yz}\\).\
p(1, 5), q(-7, 8)\
w(?, 2), x(1, -3)\
y(-8, -6), z(-3, -2)\
follow the directions below.\
(a) find the length of each segment. (decimal approximation).

Explanation:

Step1: Recall distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Calculate length of \(PQ\)

For \(P(1,5)\) and \(Q(-7,8)\), substitute into the formula:
\(x_1 = 1,y_1 = 5,x_2=-7,y_2 = 8\)
\(PQ=\sqrt{(-7 - 1)^2+(8 - 5)^2}=\sqrt{(-8)^2+3^2}=\sqrt{64 + 9}=\sqrt{73}\approx8.54\)

Step3: Calculate length of \(WX\)

For \(W(7,2)\) and \(X(1,-3)\), substitute:
\(x_1 = 7,y_1 = 2,x_2 = 1,y_2=-3\)
\(WX=\sqrt{(1 - 7)^2+(-3 - 2)^2}=\sqrt{(-6)^2+(-5)^2}=\sqrt{36 + 25}=\sqrt{61}\approx7.81\)

Step4: Calculate length of \(YZ\)

For \(Y(-8,-6)\) and \(Z(-3,-2)\), substitute:
\(x_1=-8,y_1=-6,x_2=-3,y_2=-2\)
\(YZ=\sqrt{(-3-(-8))^2+(-2 - (-6))^2}=\sqrt{(5)^2+(4)^2}=\sqrt{25 + 16}=\sqrt{41}\approx6.40\)

Answer:

Length of \(PQ=\sqrt{73}\approx8.54\), \(WX=\sqrt{61}\approx7.81\), \(YZ=\sqrt{41}\approx6.40\)