Question

translating δabc to the left 8 units and downward 3 units, we get its image δabc.
note that δabc has vertices a(6, 8), b(7, 4), and c(1, 3).
also, note that δabc has vertices a(-2, 5), b(-1, 1), and c(-7, 0).
complete the following.
(a) find each length. give exact answers (not decimal approximations).
aa = units
bb = units
cc = units
(b) find each slope. give exact answers (not decimal approximations).
slope of aa =
slope of bb =
slope of cc =
(c) choose the correct statement about the translation.
all points on the original figure moved the same distance and in the same direction.
the points on the original figure didnt all move the same distance and didnt all move in the same direction.
all points on the original figure moved the same distance, but not all points moved in the same direction.
all points on the original figure moved in the same direction, but not all points moved the same distance.

Explanation:

Part (a)

To find the length of a segment between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For a translation, the horizontal change is \(\Delta x\) and vertical change is \(\Delta y\), so the distance can also be calculated as \(\sqrt{(\Delta x)^2 + (\Delta y)^2}\). Here, the translation is 8 units left (so \(\Delta x = -8\)) and 3 units down (so \(\Delta y = -3\)).

Step 1: Calculate \(AA'\)

• Point \(A\) is \((6, 8)\) and \(A'\) is \((-2, 5)\).
• \(\Delta x = -2 - 6 = -8\), \(\Delta y = 5 - 8 = -3\).
• Using the distance formula: \(AA' = \sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}\)? Wait, no, wait. Wait, the translation is 8 left and 3 down, so the horizontal change is 8 (left) and vertical change is 3 (down). So the distance should be \(\sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73}\)? Wait, but let's check the coordinates. Wait, \(A\) is \((6,8)\), \(A'\) is \((-2,5)\). So \(6 - (-2) = 8\) (so the horizontal distance is 8 left, so \(\Delta x = -8\)), and \(8 - 5 = 3\) (vertical distance is 3 down, so \(\Delta y = -3\)). So the distance is \(\sqrt{(8)^2 + (3)^2} = \sqrt{64 + 9} = \sqrt{73}\)? Wait, but maybe I made a mistake. Wait, no, the distance formula is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). So \(x_2 - x_1 = -2 - 6 = -8\), \(y_2 - y_1 = 5 - 8 = -3\). Then \((-8)^2 = 64\), \((-3)^2 = 9\), so sum is 73, square root of 73. Wait, but let's check \(BB'\).

Step 2: Calculate \(BB'\)

• Point \(B\) is \((7, 4)\) and \(B'\) is \((-1, 1)\).
• \(\Delta x = -1 - 7 = -8\), \(\Delta y = 1 - 4 = -3\).
• Distance: \(BB' = \sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}\)? Wait, no, that can't be. Wait, no, the translation is 8 left and 3 down, so the horizontal change is 8 (left) and vertical change is 3 (down). So the distance should be \(\sqrt{8^2 + 3^2} = \sqrt{73}\). Wait, but let's check \(CC'\).

Step 3: Calculate \(CC'\)

• Point \(C\) is \((1, 3)\) and \(C'\) is \((-7, 0)\).
• \(\Delta x = -7 - 1 = -8\), \(\Delta y = 0 - 3 = -3\).
• Distance: \(CC' = \sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}\). Wait, but that seems consistent. Wait, but maybe I messed up the direction. Wait, left is negative x, down is negative y, but the distance is the same as the magnitude of the displacement vector. So the displacement vector is \((-8, -3)\), so the length is \(\sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}\). So all three distances \(AA'\), \(BB'\), \(CC'\) should be \(\sqrt{73}\)? Wait, but let's check with the coordinates. For \(A\) to \(A'\): \(x\) changes from 6 to -2: that's a change of -8 (8 units left), \(y\) changes from 8 to 5: change of -3 (3 units down). So the distance between them is \(\sqrt{(8)^2 + (3)^2} = \sqrt{73}\). Yes, because distance is absolute. So regardless of direction, the distance is the square root of (horizontal change squared plus vertical change squared). So all three segments \(AA'\), \(BB'\), \(CC'\) have length \(\sqrt{73}\). Wait, but let's confirm with the coordinates:

For \(AA'\):
\(x_1 = 6\), \(x_2 = -2\), so \(x_2 - x_1 = -8\)
\(y_1 = 8\), \(y_2 = 5\), so \(y_2 - y_1 = -3\)
Distance: \(\sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}\)

For \(BB'\):
\(x_1 = 7\), \(x_2 = -1\), so \(x_2 - x_1 = -8\)
\(y_1 = 4\), \(y_2 = 1\), so \(y_2 - y_1 = -3\)
Distance: \(\sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}\)

For \(CC'\):
\(x_1 = 1\), \(x_2 = -7\), so \(x_2 - x_1 = -8\)
\(y_1 = 3\), \(y_2 = 0\), so \(y_2 - y_1 = -3\)
Distance: \(\sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73}\)

So all thr…

The slope of a line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For a translation, the slope should be the same for all corresponding segments because translation is a rigid transformation that preserves direction (so the slope, which is the ratio of vertical change to horizontal change, should be the same).

Step 1: Calculate the slope of \(AA'\)

• Point \(A\) is \((6, 8)\) and \(A'\) is \((-2, 5)\).
• Slope \(m = \frac{5 - 8}{-2 - 6} = \frac{-3}{-8} = \frac{3}{8}\).

Step 2: Calculate the slope of \(BB'\)

• Point \(B\) is \((7, 4)\) and \(B'\) is \((-1, 1)\).
• Slope \(m = \frac{1 - 4}{-1 - 7} = \frac{-3}{-8} = \frac{3}{8}\).

Step 3: Calculate the slope of \(CC'\)

• Point \(C\) is \((1, 3)\) and \(C'\) is \((-7, 0)\).
• Slope \(m = \frac{0 - 3}{-7 - 1} = \frac{-3}{-8} = \frac{3}{8}\).

Part (c)

Translation is a rigid transformation where every point of the original figure is moved the same distance (the length of the translation vector) and in the same direction (the direction of the translation vector). So all points move the same distance and same direction.

Final Answers:

(a)

• \(AA' = \sqrt{73}\) units
• \(BB' = \sqrt{73}\) units
• \(CC' = \sqrt{73}\) units

(b)

• Slope of \(AA' = \frac{3}{8}\)
• Slope of \(BB' = \frac{3}{8}\)
• Slope of \(CC' = \frac{3}{8}\)

(c)

The correct statement is: "All points on the original figure moved the same distance and in the same direction."

Final Answers:

(a) \(AA' = \boxed{\sqrt{73}}\), \(BB' = \boxed{\sqrt{73}}\), \(CC' = \boxed{\sqrt{73}}\)

(b) Slope of \(AA' = \boxed{\frac{3}{8}}\), Slope of \(BB' = \boxed{\frac{3}{8}}\), Slope of \(CC' = \boxed{\frac{3}{8}}\)

(c) The correct option is: "All points on the original figure moved the same distance and in the same direction." (the first option)

Answer:

Translation is a rigid transformation where every point of the original figure is moved the same distance (the length of the translation vector) and in the same direction (the direction of the translation vector). So all points move the same distance and same direction.

Final Answers:

(a)

• \(AA' = \sqrt{73}\) units
• \(BB' = \sqrt{73}\) units
• \(CC' = \sqrt{73}\) units

(b)

• Slope of \(AA' = \frac{3}{8}\)
• Slope of \(BB' = \frac{3}{8}\)
• Slope of \(CC' = \frac{3}{8}\)

(c)

The correct statement is: "All points on the original figure moved the same distance and in the same direction."

Final Answers:

(a) \(AA' = \boxed{\sqrt{73}}\), \(BB' = \boxed{\sqrt{73}}\), \(CC' = \boxed{\sqrt{73}}\)

(b) Slope of \(AA' = \boxed{\frac{3}{8}}\), Slope of \(BB' = \boxed{\frac{3}{8}}\), Slope of \(CC' = \boxed{\frac{3}{8}}\)

(c) The correct option is: "All points on the original figure moved the same distance and in the same direction." (the first option)