Question

reflecting ( \triangle fgh ) across the horizontal line ( y = -2 ), we get its image ( \triangle fgh ). suppose ( overline{ff} ), ( overline{gg} ), and ( overline{hh} ) intersect the line of reflection at ( p ), ( q ), and ( r ) as shown below. complete the following. (a) suppose each pair of segments below have the same length. find each length. ( fp = fp=square ) units ( gq = gq=square ) units ( hr = hr=square ) units (b) select all that apply. ( overline{ff} ), ( overline{gg} ), and ( overline{hh} ) are each parallel to the line of reflection. ( overline{ff} ) is perpendicular to ( overline{gg} ), and ( overline{ff} ) is parallel to ( overline{hh} ). ( overline{ff} ), ( overline{gg} ), and ( overline{hh} ) 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. the line of reflection is the perpendicular bisector of each segment joining a point and its image. the line of reflection is parallel to each segment joining a point and its image. the line of reflection is perpendicular to each side of the original and final figures.

Explanation:

Step1: Calculate \(FP = F'P\)

Count the vertical distance from \(F\) (or \(F'\)) to the line \(y=-2\).
From the graph, the vertical distance from \(F\) to \(y = - 2\) is \(4\) units. So \(FP=F'P = 4\) units.

Step2: Calculate \(GQ = G'Q\)

Count the vertical distance from \(G\) (or \(G'\)) to the line \(y=-2\).
From the graph, the vertical distance from \(G\) to \(y=-2\) is \(6\) units. So \(GQ = G'Q=6\) units.

Step3: Calculate \(HR = H'R\)

Count the vertical distance from \(H\) (or \(H'\)) to the line \(y=-2\).
From the graph, the vertical distance from \(H\) to \(y=-2\) is \(2\) units. So \(HR=H'R = 2\) units.

Step4: Analyze part (b)

When reflecting a point across a horizontal line \(y = k\), the line segment joining a point and its image is perpendicular to the line of reflection. So \(\overline{FF'}\), \(\overline{GG'}\), and \(\overline{HH'}\) are each perpendicular to the line of reflection \(y=-2\).

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) \(FP = F'P=4\) units, \(GQ = G'Q = 6\) units, \(HR=H'R = 2\) units.
(b) \(\overline{FF'}\), \(\overline{GG'}\), and \(\overline{HH'}\) 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.