Question

on a regional map, four cities and a library are shown with the roads that connect them. which statement is not true? the distance from city a to the library is the same as the distance from the library to city d. the distance from city a to city b is the same as the distance from city c to city d. the distance from city c to city b is the same as the distance from city a to city d. the distance from city b to the library is the same as the distance from the library to city c.

Explanation:

Step1: Determine coordinates

First, identify the coordinates of each point:

• City A: \((1, 8)\)
• City B: \((7, 8)\)
• City C: \((3, 2)\)
• City D: \((9, 2)\)
• Library: \((5, 5)\)

Step2: Calculate distances using distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) or observe horizontal/vertical distances.

For "Distance from A to library vs library to D":

• A to Library: \(x\)-difference: \(5 - 1 = 4\), \(y\)-difference: \(5 - 8 = -3\). Distance: \(\sqrt{4^2 + (-3)^2} = 5\)
• Library to D: \(x\)-difference: \(9 - 5 = 4\), \(y\)-difference: \(2 - 5 = -3\). Distance: \(\sqrt{4^2 + (-3)^2} = 5\). So this is true.

For "A to B vs C to D":

• A to B: Horizontal distance (same \(y\)-coordinate). \(7 - 1 = 6\)
• C to D: Horizontal distance (same \(y\)-coordinate). \(9 - 3 = 6\). So this is true.

For "C to B vs A to D":

• C to B: \(x\)-difference: \(7 - 3 = 4\), \(y\)-difference: \(8 - 2 = 6\). Distance: \(\sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}\)
• A to D: \(x\)-difference: \(9 - 1 = 8\), \(y\)-difference: \(2 - 8 = -6\). Distance: \(\sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\). \(\sqrt{52}

eq 10\), so this is NOT true.

For "B to library vs library to C":

• B to Library: \(x\)-difference: \(5 - 7 = -2\), \(y\)-difference: \(5 - 8 = -3\). Distance: \(\sqrt{(-2)^2 + (-3)^2} = \sqrt{13}\)
• Library to C: \(x\)-difference: \(3 - 5 = -2\), \(y\)-difference: \(2 - 5 = -3\). Distance: \(\sqrt{(-2)^2 + (-3)^2} = \sqrt{13}\). So this is true.

Answer:

The distance from city \( C \) to city \( B \) is the same as the distance from city \( A \) to city \( D \) (the third option)