Geometric Constructions with

Understanding

**This is meant as a resource to the teacher! It is NOT intended to replace teaching in the

classroom OR the discourse that should take place!!

Inscribing in a Circle

Inscribe a Hexagon in a Circle

We will inscribe a regular hexagon in the circle.

Inscribe means that the regular hexagon’s vertices (corners) will lie on the circle and the regular hexagon will be inside the circle.

A regular hexagon has 6 sides where all are the same length & the angles are all the same measure.

A regular hexagon is also made up of 6 equilateral triangles.

Start with any circle. Make sure you mark your center and a radius.

Inscribe a Hexagon in a Circle

Segment AB is a radius of circle A.

Inscribe a Hexagon in a Circle

Construct circle B with radius AB.

Inscribe a Hexagon in a Circle

Find the points of intersection of circles A & B.

Inscribe a Hexagon in a Circle

Segment BC and segment BD are congruent to segment BA because all are radii of circle B.

Inscribe a Hexagon in a Circle

Hide circle B. Construct Circle C with radius CB.

Inscribe a Hexagon in a Circle

Segment CE and segment CA are congruent to segment CB because all are radii of circle B.

Inscribe a Hexagon in a Circle

Hide circle C. Construct Circle E with radius EC.

Inscribe a Hexagon in a Circle

Segment EF and segment EA are congruent to segment EC because all are radii of circle E.

Inscribe a Hexagon in a Circle

Hide circle E. Construct Circle F with radius FE.

Inscribe a Hexagon in a Circle

Segment FG and segment FA are congruent to segment FE because all are radii of circle F.

Inscribe a Hexagon in a Circle

Hide circle F. Construct Circle G with radius GF.

Inscribe a Hexagon in a Circle

Segment GD and segment GA are congruent to segment GF because all are radii of circle G.

Inscribe a Hexagon in a Circle

Hide circle G.

Inscribe a Hexagon in a Circle

Segment AD is a radii of circle A and therefore it is congruent to all the other radii of circle A.

Inscribe a Hexagon in a Circle

Hide all radii of circle A and you are left with a regular hexagon inscribed in circle A.

Inscribe an Equilateral Triangle in a Circle

An equilateral triangle has all 3 sides the same length & all 3 angles the same measure of 60 degrees.

Normally, we would start with a circle making sure you mark the center and a radius.

Because we have already inscribed a hexagon in a circle, we can start with that.

Inscribe an Equilateral Triangle in a Circle

Inscribe an Equilateral Triangle in a Circle

Because we have a regular hexagon, we know that that angle EFG, GDB, and BCE are congruent (120 degrees).

Inscribe an Equilateral Triangle in a Circle

We can simply connect every other point on the circle.

Inscribe an Equilateral Triangle in a Circle

By SAS congruence we have congruent triangles EFG, GDB, and BCE.

Inscribe an Equilateral Triangle in a Circle

By CPCTC, segments EG, GB, and BE are congruent.

Inscribe an Equilateral Triangle in a Circle

Hide the hexagon and we are left with an equilateral triangle.

Inscribe a Square in a Circle

A square has all 4 sides the same length & all 4 angles the same measure of 90 degrees.

Start with any circle. Make sure you mark your center and a diameter.

Inscribe a Square in a CircleSegment BC is a diameter of circle A.

Inscribe a Square in a CircleConstruct the perpendicular bisector of diameter BC.

Inscribe a Square in a CircleSegments AB, AD, AC, & AE are all congruent because they are all radii of circle A. Angles BAD, DAC, CAE, & EAB are all congruent because they are all 90 degrees.

Inscribe a Square in a CircleWe can construct segments BD, DC, CE, & EB.

Inscribe a Square in a CircleWe now have 4 congruent isosceles triangles – BAD, DAC, CAE, & EAB by SAS congruence.

Inscribe a Square in a CircleBy CPCTC, segments BD, DC, CE, & EB are congruent. Now, we know we have a rhombus inscribed in our circle. To prove it is a square, we have to prove that the angles are 90 degrees.

Inscribe a Square in a CircleIn an isosceles triangle, the base angles are congruent. The non-base angle is 90 degrees meaning that the two base angles equally share 90 degrees. So, they are all 45 degree angles.

Inscribe a Square in a CircleEach vertex on the circle is composed of two 45 degree angles making them each 90 degrees and therefore, a square.