100 POINTS Geometric Constructions
Geometric constructions date back thousands of years to when Euclid, a Greek mathematician known as the “Father of Geometry,” wrote the book Elements. In Elements, Euclid formulated the five postulates that form the base for Euclidean geometry. To create all the figures and diagrams, Euclid used construction techniques extensively. A compass and straightedge are used to create constructions. A compass is used to draw circles or arcs and a straightedge is used to draw straight lines.

As you complete the task, keep these questions in mind:
How do you perform constructions related to circles? What theorems and explanations can be used to justify these constructions?
In this task, you will apply what you have learned in this lesson to answer these questions.
Directions
Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work where indicated, including inserting images of constructions created using the tool. If you are unable to take and insert screenshots of the construction tool, print this activity sheet and create the constructions by hand using a compass and straightedge.
In addition to the answers you determine, you will be graded based on the work you show, or your solution process. So, be sure to show all your work and answer each question as you complete the task. Type all your work into this document so you can submit it to your teacher for a grade. You will be given partial credit based on the work you show and the completeness and accuracy of your explanations.
Your teacher will give you further directions about how to submit your work. You may be asked to upload the document, e-mail it to your teacher, or print it and hand in a hard copy.
Now, let’s get started!

Step 1: Construct a circle through three points not on a line.
⦁ Points D, E, and F are not in a line. To construct a circle through points D, E, and F, begin by drawing line segments and . Then construct the perpendicular bisectors of and , and name the point of intersection of the perpendicular bisectors O. How do you know that point O is the center of the circle that passes through the three points? (10 points)

Step 2: Construct regular polygons inscribed in a circle.
⦁ While constructing an equilateral triangle or a regular hexagon inscribed in a circle, you may have noticed that several smaller equilateral triangles are formed, like PQR shown in the figure below. Explain why PQR is an equilateral triangle. (5 points)

⦁ The completed construction of a regular hexagon is shown below. Explain why ACF is a 30º-60º-90º triangle. (10 points)

⦁ If you are given a circle with center C, how do you locate the vertices of a square inscribed in circle C? (5 points)

Step 3: Construct tangent lines to a circle.
⦁ is a diameter of circle K. If tangents to circle K are constructed through points L and J, what relationship would exist between the two tangents? Explain. (5 points)

⦁ The construction of a tangent to a circle given a point outside the circle can be justified using the second corollary to the inscribed angle theorem. An alternative proof of this construction is shown below. Complete the proof. (5 points)
Given: Circle C is constructed so that CD = DE = AD; is a radius of circle C.
Prove: is tangent to circle C.

Statements Reasons
⦁ Circle C is constructed so that CD = DE = AD;
is a radius of circle C. ⦁ Given
⦁ ⦁ Definition of congruence
⦁ ACD is an isosceles triangle;
ADE is an isosceles triangle. ⦁
⦁ ; ⦁
⦁ ⦁ Isosceles triangle theorem
⦁ ; ⦁ Definition of congruence
⦁ ; ⦁ Substitution property
⦁ ; ⦁ Addition
⦁ ;

⦁
⦁ and are a linear pair. ⦁
⦁ ⦁ Linear pair postulate
⦁ ⦁ Definition of supplementary angles
⦁ 1 ⦁ Substitution property
⦁ ⦁ Addition
⦁ ⦁ Subtraction property
⦁ ⦁
⦁ ⦁ Angle addition postulate
⦁ ⦁ Substitution property
⦁ is a right angle. ⦁ Definition of right angle
⦁ ⦁ Definition of perpendicular
⦁ is tangent to circle C. ⦁