Question

reflecting ( \triangle abc ) across the vertical line ( x = 1 ), we get its image ( \triangle abc ). suppose ( overline{aa} ), ( overline{bb} ), and ( overline{cc} ) intersect the line of reflection at ( x ), ( y ), and ( z ) as shown below. complete the following. (a) suppose each pair of segments below have the same length. find each length. ( a x = ax=square ) units ( b y = by=square ) units ( c z = cz=square ) units (b) select all that apply. ( overline{aa} ), ( overline{bb} ), and ( overline{cc} ) are each parallel to the line of reflection. ( overline{aa} ), ( overline{bb} ), and ( overline{cc} ) are parallel to each other. ( overline{aa} ), ( overline{bb} ), and ( overline{cc} ) are each perpendicular to the line of reflection. none of the above (c) choose the correct statement about the reflection. the line of reflection is neither parallel nor perpendicular to each segment joining a point and its image. each side of the original figure is parallel to its image. the line of reflection is parallel to each segment joining a point and its image. the line of reflection is the perpendicular bisector of each segment joining a point and its image.

Explanation:

Step1: Calculate \(AX = A'X\)

The line of reflection is \(x = 1\). For point \(A\) (let's assume \(A\) has \(x\) - coordinate \(x_A=-2\)). The distance from \(A\) to the line \(x = 1\) is \(|1-(-2)|=3\) units. So \(AX=A'X = 3\) units.

Step2: Calculate \(BY = B'Y\)

For point \(B\) (assume \(B\) has \(x\) - coordinate \(x_B = 8\)). The distance from \(B\) to the line \(x = 1\) is \(|8 - 1|=7\) units. So \(BY=B'Y = 7\) units.

Step3: Calculate \(CZ = C'Z\)

For point \(C\) (assume \(C\) has \(x\) - coordinate \(x_C=-4\)). The distance from \(C\) to the line \(x = 1\) is \(|1-(-4)|=5\) units. So \(CZ=C'Z = 5\) units.

Step4: Analyze part (b)

• The line of reflection \(x = 1\) is a vertical line. A segment joining a point and its reflection (e.g., \(\overline{AA'}\), \(\overline{BB'}\), \(\overline{CC'}\)) is horizontal. Horizontal lines are perpendicular to vertical lines. So \(\overline{AA'}\), \(\overline{BB'}\), and \(\overline{CC'}\) are each perpendicular to the line of reflection.

Step5: Analyze part (c)

By the property of reflection, the line of reflection is the perpendicular bisector of each segment joining a point and its image.

Answer:

(a) \(AX = A'X=3\) units, \(BY = B'Y = 7\) units, \(CZ=C'Z = 5\) units.
(b) \(\overline{AA'}\), \(\overline{BB'}\), and \(\overline{CC'}\) are each perpendicular to the line of reflection.
(c) The line of reflection is the perpendicular bisector of each segment joining a point and its image.