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.

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.


Segment AB is a radius of circle A.


Construct circle B with radius AB.


Find the points of intersection of circles A & B.


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


Hide circle B. Construct Circle C with radius CB.


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


Hide circle C. Construct Circle E with radius EC.


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


Hide circle E. Construct Circle F with radius FE.


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


Hide circle F. Construct Circle G with radius GF.


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


Hide circle G.


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


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.



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


We can simply connect every other point on the circle.


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


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


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.

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.

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