Solve the linear equation 2(3x + 2) = 2x - 1 + x. State the property that justifies your first step and why you chose it.

D

Step-by-step explanation:

Given: a piece of paper

Construct: a regular hexagon inscribed in a circle

STEPS:

1. Place your compass point on the paper and draw a circle. (Keep this compass span!)

2. Place a dot, labeled P, anywhere on the circumference of the circle to act as a starting point.

3. Without changing the span on the compass, place the compass point on P and swing a small arc crossing the circumference of the circle.

4. Without changing the span on the compass, move the compass point to the intersection of the previous arc and the circumference and make another small arc on the circumference of the circle.

5. Keep repeating this process of "stepping" around the circle until you return to point P.

6. Starting at P, connect to each arc on the circle forming the regular hexagon.

Hex1

Full circle view:

Hex2

dashdivider

Proof of Construction: PA = AB = BC = CD = DE since these lengths represent copies of the radius of circle O. But how do we know for sure that the last length, EP, coincided exactly with point P? Is EP actually the same length as the other copied radii?

ΔDOE is an equilateral triangle since it has 3 sides of equal length (DO and OE are radii lengths and DE is a copy of this radii length). In similar fashion, ΔCOD, ΔBOC. ΔAOB and ΔPOA are also equilateral triangles. Since the interior angles of an equilateral triangle each contain 60º, m∠COD = m∠BOC = m∠AOB = m∠POA = m∠DOE = 60º. Since all of the central angles (surrounding a point) must add to 360º, we know the m∠POE = 60º (360º - 300º = 60º).

Since Hexsideswe have ΔDOE congruent ΔDOE by SAS. By CPCTC, deep and we know that the last copy of the radii coincides with point P making the hexagon truly inscribed. Hexagon PABCDE has all vertices on circle O, has congruent interior angles (each equal 120º) and has all sides congruent. PABCDE is an inscribed regular hexagon by definition.