Question

the endpoints of two segments are given. find the length of each segment. if necessary, round your answers to the nearest tenth. (overline{ef}): (e(1, 4)), (f(5, 1)) and (overline{gh}): (g(-3, 1)), (h(1, 6)) the length of (overline{ef}) is (square). the length of (overline{gh}) is (square). tell whether the segments congruent. (circ) yes (circ) no if they are not congruent, tell which segment is longer. (circ) (overline{ef}) is longer. (circ) (overline{gh}) is longer. (circ) (overline{ef}) and (overline{gh}) are congruent.

Explanation:

Step1: Recall the 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 the length of \(\overline{EF}\)

For \(E(1, 4)\) and \(F(5, 1)\), substitute into the distance formula:
\(x_1 = 1\), \(y_1 = 4\), \(x_2 = 5\), \(y_2 = 1\)
\(EF=\sqrt{(5 - 1)^2 + (1 - 4)^2}=\sqrt{4^2 + (-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\)

Step3: Calculate the length of \(\overline{GH}\)

For \(G(-3, 1)\) and \(H(1, 6)\), substitute into the distance formula:
\(x_1 = -3\), \(y_1 = 1\), \(x_2 = 1\), \(y_2 = 6\)
\(GH=\sqrt{(1 - (-3))^2 + (6 - 1)^2}=\sqrt{(4)^2 + (5)^2}=\sqrt{16 + 25}=\sqrt{41}\approx 6.4\) (rounded to the nearest tenth)

Step4: Determine congruence and longer segment

Since \(EF = 5\) and \(GH\approx 6.4\), \(5
eq6.4\), so the segments are not congruent. And \(GH\) is longer.

Answer:

The length of \(\overline{EF}\) is \(5\). The length of \(\overline{GH}\) is \(6.4\).
Tell whether the segments congruent: No
If they are not congruent, tell which segment is longer: \(\overline{GH}\) is longer.