Step 1: Construct a circle through three points not on a line.
a) 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 DE and EF. Then construct the perpendicular bisectors of D and EF, 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)
O
D
F
E
Student Guide (continued)

Step 2: Construct regular polygons inscribed in a circle.
a) 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)
b) The completed construction of a regular hexagon is shown below. Explain why ACF is a 30º60º-90º triangle. (10 points)
E
D
B
A
C
F
M
R
Q
P
Student Guide (continued)

c) 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.
a) JL 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)
Student Guide (continued)

b) 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; CA is a radius of circle C,
CE is a diameter
of circle D.
Prove: AE is tangent to circle C.
Statements
Reasons
1. Circle C is constructed so that CD = DE = AD;CA is a radius of circle C;CE is a diameter of
circle D.
1. Given
2. CD≅ DE ≅ AD 2. Definition of congruence
3. ACD is an isosceles triangle;
ADE is an isosceles triangle.
3.
4. m∠CAD + m∠DCA + m∠ADC = 180°;
m∠DAE + m∠AED + m∠EDA = 180°
4.
5. 5. Isosceles triangle theorem
6. m∠CAD= m∠DCA; m∠DAE = m∠AED 6. Definition of congruence
7. m∠CAD + m∠CAD + m∠ADC = 180°;
m∠DAE + m∠DAE + m∠EDA = 180°
7. Substitution property
8. 2(m∠CAD) + m∠ADC = 180°;
2(m∠DAE) + m∠EDA = 180°
8. Addition
9. m∠ADC = 180° – 2(m∠CAD);
m∠EDA = 180° – 2(m∠DAE)
9.
10. ∠ADC and ∠EDA are a linear pair. 10.
A
D C
E
Student Guide (continued)

11. 11. Linear pair postulate
12. m∠ADC + m∠EDA = 180° 12. Definition of supplementary
angles
13. 180° – 2(m∠CAD) + 180° – 2(m∠DAE) = 180° 13. Substitution property
14. 360° – 2(m∠CAD) – 2(m∠DAE) = 180° 14. Addition
15. – 2(m∠CAD) – 2(m∠DAE) = −180° 15. Subtraction property
16. m∠CAD+ m∠DAE = 90° 16.
17. m∠CAD+ m∠DAE = m∠CAE 17. Angle addition postulate
18. 18. Substitution property
19. ∠CAE is a right angle. 19. Definition of right angle
20. 20. Definition of perpendicular
21. AE is tangent to circle C. 21.

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