Question

on a coordinate plane, polygon ghij translates 8 units to the left to form polygon ghij. which of the following equations is not necessarily true?

○ a. gh = gh
○ b. gg = 8 units
○ c. m∠hij = m∠hij
○ d. m∠hij = m∠hij

Explanation:

Step1: Properties of translation

Translation is a rigid transformation. In a rigid transformation, the shape and size of the figure do not change. So, the corresponding sides and corresponding angles of the pre - image and the image are equal.
For a translation of a polygon \(GH IJ\) to \(G'H'I'J'\) by \(8\) units to the left:

• Corresponding sides: \(GH = G'H'\), \(HI=H'I'\), \(IJ = I'J'\), \(JG=J'G'\)
• Corresponding angles: \(m\angle{G}=m\angle{G'}\), \(m\angle{H}=m\angle{H'}\), \(m\angle{I}=m\angle{I'}\), \(m\angle{J}=m\angle{J'}\)

Also, the distance between a point and its image (e.g., for point \(G\) and \(G'\)) is equal to the distance of translation. So, \(G'G = 8\) units.

Step2: Analyze each option

• Option A:

Since translation is a rigid transformation, \(GH = G'H'\) (corresponding sides of congruent polygons). This equation is True.

• Option B:

The translation is \(8\) units to the left. The distance between a pre - image point \(G\) and its image point \(G'\) is \(8\) units. So, \(G'G=8\) units. This equation is True.

• Option C:

\(\angle HIJ\) and \(\angle H'I'J'\) are corresponding angles of the pre - image and the image of the translated polygon. Since translation is a rigid transformation, \(m\angle HIJ=m\angle H'I'J'\). This equation is True.

• Option D:

\(\angle HI'J\) and \(\angle H'IJ'\) are not corresponding angles of the pre - image and the image of the translated polygon. There is no property of translation that guarantees \(m\angle HI'J = m\angle H'IJ'\). This equation is not necessarily True.

Answer:

D. \(m\angle HI'J = m\angle H'IJ'\)