The construction of the regular pentagon is equivalent to the construction of the point in the unit circle with x coordinate x1 = the square root of 5 - 1 divided by 4. Use this to construct a regular pentagon that has all its vertices inside the unit circle. Write a construction protocol and justify all your construction steps. How do I construct the square root of 5-1?
You must mean
√5 - 1
draw a square ABCD with sides 2 , where A and B are the endpoints of the base
Extend the length of the base to the right
Draw a vertical MN where M is the midpoint of AB and N is the midpoint of CD
You should see a right-angled triangle with legs AB = 1 and MN = 2
by Pythagoras AN = √5
Use your compass, centre at A and radius AN , draw an arc to cut the entension of AB at P
so AP = √5
and MP = √5 - 1
To construct the regular pentagon with all its vertices inside the unit circle using the given x-coordinate, you first need to know how to construct the square root of 5-1. Here's a step-by-step guide:
Construction of the square root of 5-1:
1. Draw a horizontal line and mark two points on it, say A and B.
2. With A as the center, draw a circle passing through point B. Label the intersection point with the circle as C.
3. Extend the line segment AB beyond point B and mark a point on it, labeled D, such that AD = AC.
4. Draw a perpendicular bisector of the segment AD, and let E be the point of intersection between it and AD.
5. With E as the center, draw a circle passing through point D and label the intersection point with the circle as F.
6. Draw a line segment joining point C and point F, and label the intersection point with the circle as G.
7. Draw a perpendicular line from point G to the line segment FD, and label the point of intersection as H.
8. Finally, draw a line segment from point A to point H. The length of this line segment AH is equal to the square root of 5-1.
Now, once you have the square root of 5-1, you can proceed to construct the regular pentagon with its vertices inside the unit circle.
Construction of the regular pentagon:
1. Draw a unit circle centered at the origin.
2. On the x-axis, mark the point with the x-coordinate of x1 = √(5-1) / 4.
3. Draw a perpendicular line through the marked point on the x-axis, intersecting the unit circle at points A and B.
4. Using point A as the center, draw a circle passing through point B, labeling the point of intersection with the circle as C.
5. Connect points B and C, extending the line segment past point C.
6. Bisect the line segment BC, labeling the point of intersection with the bisector as D.
7. Using point D as the center, draw a circle passing through point B and label its intersection point with the circle as E.
8. Connect points C and E.
9. Bisect the line segment CE, labeling the point of intersection with the bisector as F.
10. Draw a line segment from point C to point F.
11. Label the intersection point of the line segment CF with the unit circle as G.
12. Draw a line segment from the center of the unit circle (origin) to point G.
13. Repeat steps 11 and 12 to obtain the remaining two vertices of the pentagon, labeling them as H and I, respectively.
14. The resulting figure connecting points A, B, C, D, and E is a regular pentagon with all its vertices inside the unit circle.
By following these construction steps, you will be able to construct the regular pentagon as required.