Question

visualize the following information.
line ab || line cd, line ef ⊥ line ab, line ef ⊥ line cd
line ef intersects line ab at point p, and line cd intersects line ef at point q.
which statement is true?
a. if ef is rotated 180° clockwise about point q to become ef, then ef || ab.
b. if ab is rotated 180° clockwise about point p to become ab, then ab ⊥ cd.
c. if cd is rotated 270° clockwise about point q to become cd, then cd ⊥ ab.
d. if ab is rotated 90° clockwise about point p to become ab, then ab ⊥ ef.

Explanation:

Step1: Analyze Option A

Rotating \(EF\) \(180^{\circ}\) about \(Q\) makes \(E'F'\) coincide with \(EF\) (since rotation of \(180^{\circ}\) about a point on the line results in the line itself). But \(EF\perp AB\), so \(E'F'\) is not parallel to \(AB\).

Step2: Analyze Option B

Rotating \(AB\) \(180^{\circ}\) about \(P\) makes \(A'B'\) coincide with \(AB\) (rotation of \(180^{\circ}\) about a point on the line). Since \(AB\parallel CD\), \(A'B'\parallel CD\), not \(A'B'\perp CD\).

Step3: Analyze Option C

Rotating \(CD\) \(270^{\circ}\) clockwise about \(Q\) is equivalent to rotating \( - 90^{\circ}\) (or \(270^{\circ}\) clockwise \(=360^{\circ}-90^{\circ}\)). If \(CD\) is rotated \( - 90^{\circ}\) about \(Q\), \(C'D'\) will be parallel to \(EF\). Since \(EF\perp AB\), \(C'D'\perp AB\).

Step4: Analyze Option D

Rotating \(AB\) \(90^{\circ}\) clockwise about \(P\). Since \(AB\parallel CD\) and \(EF\perp AB\), rotating \(AB\) \(90^{\circ}\) about \(P\) will make \(A'B'\) parallel to \(CD\) (not perpendicular to \(EF\) as \(EF\) is already perpendicular to \(AB\) and \(CD\)).

Answer:

C. If \(CD\) is rotated \(270^{\circ}\) clockwise about point \(Q\) to become \(C'D'\), then \(C'D'\perp AB\)