Geometry · Unit 2
Compass and straightedge items from the released Geometry tests, 2000 through 2010. Each card names the test year and the problem number. Open the test for the figure.
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Same compass opening.
The new line goes through the crossing of the arcs.
Perpendicular from a point off the line
Open the compass at the outside point so the arc crosses the line twice. From those two crossings, swing equal arcs that meet. The perpendicular goes through the outside point and that meeting.
Perpendicular at a point on the line
Center the first arc at the point on the line so it crosses the line on both sides. From those crossings, swing equal arcs that meet. The perpendicular goes through the point and that meeting.
Perpendicular bisector
Open the compass to more than half the segment. Swing that same opening from each endpoint. The arcs meet above the segment and below it. The line through those two meetings cuts the segment in half and is perpendicular to it.
Angle bisector
Swing an arc from the vertex so it crosses both sides. From those two crossings, swing equal arcs that meet inside the angle. The bisector is the ray from the vertex through that meeting.
Copy a segment
Open the compass to the given segment. Keep that opening. On the new ray, the compass mark is the other endpoint of the congruent segment.
Copy an angle
Swing an arc from the given vertex. Swing the same arc from the endpoint of the new ray. Open the compass to the chord of the given angle, and mark that opening on the new arc. The new side goes through that mark.
The tools
A geometric construction uses a compass and a straightedge. The compass holds a length or draws an arc. The straightedge draws the line through two points. A protractor is not one of the tools.
The point is on the line, or it is off the line. The right line meets the given line at a right angle.
The given point is not on the line.
Swing an arc from that point so it crosses the line twice.
From those two crossings, swing equal arcs that meet.
The perpendicular is the line through the given point and that meeting point.
That line also passes through W.
Answer: A
The given point is on the line, so the perpendicular goes through that point.
Center the first arc at that point so it crosses the line on both sides.
From those two crossings, swing equal arcs that meet off the line.
The perpendicular passes through Y.
Answer: H
The given point is on the segment.
The perpendicular through that point meets the segment at a right angle.
That perpendicular passes through Q.
Answer: A
The given point is on the line.
Center the compass there, mark two crossings on the line, then swing equal arcs from those crossings.
The perpendicular passes through X.
Answer: B
The given point is on the line.
The perpendicular through that point is the line that meets the given line at a right angle.
That perpendicular passes through Y.
Answer: H
A rectangle has a right angle at each corner.
The new side has to be perpendicular to the bottom side.
That perpendicular meets the bottom side at X.
Answer: B
The point is not on the segment.
The perpendicular is the line through that point that meets the segment at a right angle.
That line is PW.
Answer: A
The given point is not on the line.
The perpendicular through that point crosses the line at a right angle and keeps going.
That perpendicular passes through X.
Answer: F
The given point is not on the line.
The perpendicular through that point meets the line at a right angle.
That perpendicular passes through W.
Answer: H
The line cuts the segment in half and is perpendicular to it.
The perpendicular bisector passes through the midpoint.
It is perpendicular to the segment.
X and Z are the pair on that line.
Answer: G
The line has to cut the segment in half and meet it at a right angle.
Swing equal arcs from the two endpoints. They meet above the midpoint and below the midpoint.
The line through those meetings passes through S.
Answer: C
The bisector of a segment is its perpendicular bisector.
It passes through the midpoint and is perpendicular to the segment.
That line contains EF.
Answer: D
The ray from the vertex splits the angle into two equal angles.
Swing an arc from the vertex so it crosses both sides.
From those two crossings, swing equal arcs that meet inside the angle.
The bisector is the ray from the vertex through that meeting point.
That ray passes through Y.
Answer: H
The bisector is the ray from the vertex through the meeting of the equal arcs.
That ray passes through X.
Answer: B
The bisector splits the angle into two equal angles.
It passes through the meeting of the equal arcs.
That ray passes through K.
Answer: C
Swing an arc from the vertex across both sides, then equal arcs from those crossings.
The bisector goes from the vertex through the meeting of those arcs.
That ray passes through S.
Answer: H
The bisector is the ray from the vertex through the arc meeting.
That ray passes through C.
Answer: H
Equal shares means the cut splits the angle in half.
That cut is the angle bisector.
The bisector is CX.
Answer: C
The arcs show an angle bisector: an arc from the vertex, then equal arcs that meet on the new ray.
An angle bisector makes the two angles equal.
It does not have to make the sides equal.
So m∠ABD = m∠CBD.
Answer: C
The bisector is the ray from the vertex through the meeting of the equal arcs.
That ray passes through Y.
Answer: H
Keep the compass opening. Match the given length.
Open the compass from one endpoint of the given segment to the other.
Keep that opening. Place the compass point at the start of the other ray.
The mark that matches that opening is E.
CE ≅ AB.
Answer: B
Open the compass to the given segment.
Keep that opening and mark it from the shared endpoint along the other ray.
AZ ≅ AB.
Answer: J
Open the compass to the given segment.
The same opening matches a segment on the other line.
CN ≅ AB.
Answer: D
Open the compass to the given segment.
Compare that opening with the segments from the same endpoint.
AD ≅ AB.
Answer: F
Open the compass to the given segment.
Keep that opening and mark it from the endpoint of the ray.
PR ≅ BC.
Answer: G
Read the arcs, then name the construction they make.
An arc is swung from the given vertex, and the same arc is swung from the endpoint of a new ray.
A second mark copies the opening of the given angle.
That is an angle congruent to a given angle.
Answer: D
The point is off the line, so the first arc starts there and crosses the line twice.
Equal arcs from those two crossings meet off the line.
The perpendicular joins the outside point to that meeting point.
That is drawing D.
Answer: D
The arc from the given angle is copied onto a new ray.
The second mark copies the opening, and the new side goes through that mark.
That is an angle congruent to a given angle.
Answer: B
An arc from the vertex crosses both sides.
Equal arcs from those crossings meet, and the ray from the vertex goes through that meeting.
That is the bisector of the given angle.
Answer: C
The center point is on the line. This is a perpendicular at a point on the line.
The first arc is centered at that point and crosses the line on both sides.
The later arcs are centered where that first arc meets the line.
Arc 1 is first.
Answer: A
An arc from the vertex crosses both sides.
Equal arcs meet, and the ray from the vertex goes through that meeting.
That is the bisector of the given angle.
Answer: G
The same compass opening is used from each endpoint of the segment.
The arcs meet above the segment and below the segment.
The line through those two meetings is the perpendicular bisector.
Answer: D
Decide what a construction uses, and what it does not use.
A construction uses a compass and a straightedge.
A protractor measures degrees. It is not a construction tool.
A ruler used to measure a length is not the straightedge in a construction. The straightedge only draws the line.
Answer: false
The compass draws the arcs and keeps an opening.
The straightedge joins two points.
Answer: true
That is measuring and drawing, not a construction.
The construction copies the opening with the compass.
Answer: false
More than half lets the arcs meet.
The same opening is used from each endpoint.
Answer: true
The compass holds the length.
The straightedge draws the line. It does not measure.
Answer: false
That arc crosses both sides of the given angle.
The same opening is then used at the endpoint of the new ray.
Answer: true
Need help seeing which arc comes first?
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