This diagram is a straightedge and compass construction. Select all true statements

Question

Diagram: https://assets.learnosity.com/organisations/154/10b22f49-012d-4232-89f5-6ba828fa8387.png

A. The line EF is the bisector of angle BAC

B. Line EF is the perpendicular bisector of segment BA

C. Line EF is the perpendicular bisector of segment AC

D. Line EF is the perpendicular bisector of segment BD

E. Line EF is parallel to line CD

Answer 1

Hi, Candi.

Let's look at each answer in turn:



A: TRUE

Reason: The larger circle has center A. The two smaller circles have centers B and D, which lie on the larger circle. These two smaller circles are of equal radii (radiuses) and intersect line EF at points E and F, and they intersect at each other's centers (B and D). These are the only conditions under which line EF could bisect ∠BAC, which indeed it does.

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B. FALSE

Reason: A perpendicular bisector is a ray, line, or segment that a) bisects another b) ray, line, or segment at c) 90°.

While b) is true, as BA is indeed a segment, neither a) nor c) is true: For EF to bisect BA, EF would have to intersect BA at BA's midpoint; instead, it bisects BA at point A, which is BA's endpoint.

Nor is a 90° angle formed at point A, as EF would have to lie outside ∠BAC.

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C. FALSE

Reason: See the anwer for B.

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D. FALSE

Reason: Perpendicular bisectors bisect rays, lines, or segments (as mentioned in answer B. BD is not a segment because segments are straight; BD is an arc of the larger circle, and arcs are not straight. (BD was incorrectly labeled "segment" in the question -- perhaps to make it a 'trick question.')

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E. FALSE

Reason: Parallel lines are equidistant from each other and as such never intersect. Line EF intersects line CD at point A. (Remember: line EF is the same line as line CD.)

I hope this helps.

Please let me know if have any questions :)