Unit 13: Constructions
Every classic compass-and-straightedge construction, drawn step by step, and why the arcs alone are enough to guarantee exactness.
6 topics · 21 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Constructing Congruent Segments and Angles
- How it worksHow does opening a compass to one segment's length let you mark off an identical segment somewhere else?
- How it worksHow do you copy an angle using only arcs, without ever measuring it with a protractor?
- Worked exampleShow how to construct a segment three times as long as a given 4 centimeter segment by copying it end to end.
- Common mistakeWhy doesn't a compass-and-straightedge construction allow you to just measure with a ruler and mark it?
Your turn1 of 2
A segment measures 7 centimeters. Using compass-and-straightedge construction, it's copied end to end 5 times. Find the total length constructed.Constructing a Perpendicular Bisector
- How it worksHow do the two intersection points of equal-radius arcs from both endpoints define the perpendicular bisector line?
- Worked exampleShow how to construct the perpendicular bisector of a 6 centimeter segment using compass arcs of radius 5.
- Common mistakeWhy doesn't the perpendicular bisector construction work if the two arcs are drawn with different radii?
- Going furtherWhat happens to the perpendicular bisector construction when it's used on a segment that's part of a triangle's side?
Your turn1 of 2
Why doesn't the perpendicular bisector construction work if the two arcs, one from each endpoint, are drawn with different radii?Constructing an Angle Bisector
- How it worksHow does finding a point equidistant from both sides of an angle guarantee it lies on the bisector?
- Worked exampleShow how to construct the bisector of a 70 degree angle using compass arcs.
- Ties togetherHow does the angle bisector construction use the same equal-arcs idea as the perpendicular bisector construction?
- Going furtherWhat happens to an angle bisector construction when it's performed on a straight 180 degree angle?
Your turn1 of 2
An angle measures 121 degrees. Using compass arcs, its bisector is constructed. Find the measure of each of the two resulting angles.Constructing Parallel and Perpendicular Lines
- How it worksHow does constructing a perpendicular through a point off a line rely on two intersecting arcs?
- Worked exampleShow how to construct a line through a given point parallel to a given line using an angle copy.
- Going furtherWhat happens to the construction steps for a perpendicular line when the given point sits directly on the line instead of off it?
Your turn1 of 2
To construct a perpendicular to a line through a point NOT on the line, an arc is swung from that point, crossing the line at two spots. Then two more equal arcs are swung from those two crossing points, on the opposite side of the line. What do those last two arcs locate?Constructing an Equilateral Triangle and a Regular Hexagon
- How it worksHow does swinging one arc from each endpoint of a segment locate the third vertex of an equilateral triangle?
- Worked exampleShow how to construct a regular hexagon inscribed in a circle of radius 4 using only the circle's own radius.
- Common mistakeWhy doesn't stepping a compass around a circle at a different radius still produce a regular hexagon?
Your turn1 of 2
A regular hexagon is inscribed in a circle of radius 15, using the circle's own radius as the compass setting to step around it. Find the length of each side of the hexagon.Constructing a Triangle's Inscribed and Circumscribed Circles
- How it worksHow does dropping a perpendicular from the incenter to any side find the exact radius of the inscribed circle?
- Worked exampleShow how to construct the circumscribed circle of a triangle using the perpendicular bisectors of its sides.
- Ties togetherHow do the circumcenter and incenter constructions both reuse the same bisector techniques from earlier in the course?
Your turn1 of 2
A triangle's inscribed circle has radius 11, and one side of the triangle has length 20. Find the area of the smaller triangle formed by that side and the two radii drawn from the incenter to its endpoints.