Chapter 7: Quadrilaterals
In this packet, you will learn various methods for constructing special quadrilaterals, and you will explore various properties of special quadrilaterals.
#1: Construct an Isosceles Triangle and a Kite
Use the instructions and example on the front page of the packet to guide your construction.
1) Using the tools & strategies you have been using on previous assignments (circles, compass, etc.), build kite KITE so that
- MN = KI = TI
- KE = TE
- KE > KI
#1: Kite + its Diagonals
#2: Two Special Kites: A Rhombus and a Concave Kite
Use the instructions and example on p. 11 in the packet to guide your construction.
If you make a kite with four congruent sides, you will end up making a rhombus.
Construct rhombus RHOM by doing the following:
1) Construct an isosceles triangle above "RO" (2 congruent circles, intersection, etc.).
2) Construct another isosceles triangle - this time with the same lengths - below "RO."
3) Label the top intersection "H" and the bottom intersection "M." RHOM is a rhombus, because each side is a radius of congruent circles.
4) Draw the other diagonal, "HM."
Keep your circles visible, but lighter/thinner, as evidence of your work.
#2a: Rhombus RHOM
#2b. Concave Kite
1. Using any radius, construct an isosceles triangle above the given base.
2. With a different radius, construct another isosceles triangle - also above the given base.
3. Label the new corner points I and E.
4. Draw the diagonal between I and E.
Keep your circles visible, but lighter/thinner, as evidence of your work.
#2b: Concave Kite
The diagonals of KITE are between I & E, and between K & T. Compare the concave kite above with the convex kite in #1a. How are the diagonals of a concave kite different from those of a convex kite?
#3: Useful Properties of Kites Use the diagram on page 13 of the packet. A kite is formed from two isosceles triangles with a common base. These triangles may be tipped sideways, as in the picture on the right. The common base, "KT," is one diagonal of a kite. The other diagonal is called the main diagonal ("EI" in the diagram). The main diagonal has several properties that are used extensively in construction. See if you can discover them. a. What type of angle is angle 4: acute, right, or obtuse?
b. Which length is longer: KM, TM, or are they always equal?
c. Which angle is wider: angle 1, angle 2, or are they always equal?
d. Which angle is wider: angle 5, angle 6, or are they always equal?
Key properties to notice:
1. The main diagonal of a kite is the perpendicular bisector of the other diagonal!
(Why? You can prove that both endpoints of the main diagonal - E & I - must lie on the perpendicular bisector of the other diagonal, because they are both equidistant from K & T [see perpendicular bisector theorem].)
2. This means that the diagonals of a kite are always perpendicular, and
3. The main diagonal bisects the other diagonal.
4. The main diagonal is the angle bisector of each angle (this is prove-able with triangle congruence + CPCTC).
#4: Find a Segment's Midpoint Using the Kite Construction
Use the instructions and example on p. 14 of the packet to guide your construction.
1) As in #1a, follow the process of making an isosceles triangle above the given segment. (Make a circle centered at A. Make a circle with the same radius, centered at B.)
2) Label the intersection point above as "C," but don't add the segments connecting it to A or B.
3) Repeat these steps below the segment, but with a different radius.
4) Label the intersection point below the segment as "D."
5) Make a line that goes through C and D. This is the perpendicular bisector of the original segment (prove-able through triangle congruence + CPCTC!).
6) Place a point where the lines intersect and label it "M." This is the midpoint of the segment.
Keep your circles visible, but lighter/thinner, as evidence of your work.
#4: Midpoint Construction
Notice in #4 that ACBD would create a kite if you connected the dots. In a kite, the main diagonal ("CD" in our example) is the perpendicular bisector of the other diagonal. This is why you can use it to find a midpoint. Explain why CD must be the perpendicular bisector of AB. (Hint: by construction, CA = CB and DA = DB. What deductions can you make from that information?) (ignore my notation errors - bar above vs. not above isn't possible in this program.)
#5: Construct a Square from its Diagonal
Use the instructions and example on p. 15 of the packet to guide your construction.
If you construct a rhombus, BUT you make it equiangular, you will produce a square.
Conveniently for us, the four corners of a square lie on a circle (see diagram on p. 15).
Using "SU" below, construct the original square that has "SU" as its diagonal.
1) Construct the perpendicular bisector of "SU" (follow the midpoint steps from section #4 above).
2) Label midpoint of "SU", "R."
3) Make a circle with center R that goes through S.
4) Label where the circle crosses the perpendicular bisector, "Q" and "A."
5) Add segments "SQ," "UQ," "SA," and "UA." SQUA should now be a square!
Check your work by moving around the original segment, "SU." The square should adjust accordingly while remaining a square.
Keep your circles visible, but lighter/thinner, as evidence of your work.
#5: Construct a Square from its Diagonal
#6: Bisect an Angle
Use the instructions and example on p. 16 of the packet to guide your construction.
This is a review from the Chapter 6 assignment. Remember, to bisect an angle with construction tools, you can:
1) Construct a circle centered at B. Label where it intersects the two sides of the angle as "A" and "C".
2) Construct a circle centered at A.
3) With the COMPASS tool, use the same radius as circle A to construct a circle centered at C.
4) Label the intersection of circle A & circle C (the intersection that is further away from B) as point D.
5) Construct a ray from B through D. This is the angle bisector of angle ABC.
Keep your circles visible, but lighter/thinner, as evidence of your work.
#6a: Bisect an Angle
Check your work: use the purple points to click and move around the angle.
The ray should always bisect the angle, no matter how wide or narrow you make the angle.
NOTICE that if you connected the points, BADC would be a kite!
Now, use the same procedure from #6a to bisect right angle P below.
#6b: Bisect a Right Angle
#7: Construct a Regular Octagon
Use the instructions and example on p. 17 of the packet to guide your construction.
^^^^ DEFINITELY make sure you're looking at the diagram on this page as you go.
A regular octagon has eight congruent sides and eight congruent angles. Like a square, all 8 vertices of a regular octagon lie on a circle.
Follow the steps to construct a regular octagon below:
1) Construct theperpendicular bisector of "AE."
2) Label the midpoint of "AE," "M."
3) Place a circle with center M, passing through A.
4) Label the points where the circle intersects the perpendicular bisector: "C" and "G."
NOTICE: M is like the "origin" where four right angles all meet. In the next steps, you will construct the angle bisector of each of these right angles.
5) Place a circle centered at A, through M.
6) Place a circle centered at C, through M.
7) Place a circle centered at E, through M.
8) Place a circle centered at G, through M.
9) There should now be four intersection points, one in each "quadrant" (top left, top right, bottom left, bottom right). For each intersection, draw a ray going through it, starting from M. These rays should now bisect the original four right angles.
10) Label where the circle with center M intersects these rays - "B" "D" "F" and "H" - so that all the points are in alphabetical order around the circle.
11) Add segments connecting the eight points on the circle to form a regular octagon ABCDEFGH!
Keep your circles visible, but lighter/thinner, as evidence of your work.
#7: Construct a Regular Octagon
#8: Construct a Regular Dodecagon (Clock)
Look back at the construction of the regular octagon. Ignore the angle bisectors, ignore the points B, D, F, and H, and ignore the octagon.
Can you see how the arcs you created formed four petals of a flower?
Can you see how to draw those petals using just four circles?
These circles, together with the vertical & horizontal segments, cross the circle with center M in 12 equally spaced points. These points will thus form the vertices of a regular dodecagon.
1) On the segment below, recreate the construction of the four "petals" from the octagon, using the perpendicular bisector, circle M, and the four "outer" circles. Omit the angle bisectors, and omit the octagon.
2) Locate the 12 equidistant points on the circle, and use them to construct a regular dodecagon - an equilateral and equiangular 12-gon.
3) Using the text-box feature, decorate your dodecagon with the numbers of a clock!
Keep your circles visible, but lighter/thinner, as evidence of your work.
#8: Construct a Regular Dodecagon (Clock)
#9: Join the Midpoints of Any Quadrilateral
Use the diagram in p. 59 of the packet to help you.
The shape formed by the midpoints of any quadrilateral has an interesting property. Construct such a shape to help you figure out what this property must be.
1) Notice that ABCD below is just a general, malleable quadrilateral. You can click and move the points around to make any type of quadrilateral that you want. This will be important!
2) Make a circle centered at A with a BIG radius (more than half the length of the longest side).
3) Use the COMPASS tool to make a circle with the same radius, centered at B.
4) Use the INTERSECT tool to place points at the two intersection points, then draw the segment connecting these two points.
5) Place a point "M" where the segment intersects side "AB." This is the midpoint!
6) Repeat steps 3-5 using a circle with the same radius centered at C, to identify and label "N," the midpoint of "BC."
7) Repeat steps 3-5 using a circle with the same radius centered at D, to identify and label "O"" the midpoint of "CD."
8) Repeat steps 4-5 to identify and label "P," the midpoint of "AD."
9) Connect the midpoints to form quadrilateral MNOP.
Keep your circles visible, but lighter/thinner, as evidence of your work.
#9: The Midpoint Quadrilateral
Now, with the "MOVE" button, click and drag ABCD to manipulate the shapes. a) What special type of quadrilateral does MNOP continue to be, no matter how you stretch, bend, or rotate ABCD?
To verify this conjecture, draw the diagonal connecting A & C, then answer the following questions: b) Which segment is the midsegment of triangle ABC?
c) Which segment is the midsegment of triangle ADC?
d) Why will MN always be parallel to OP?
e) Why will MN = OP always?
f) How do you know that MNOP will always be the special type of quadrilateral you said it would be?
#10: Join the Midpoints of a Rectangle and Peer Into the Infinitesimal!
Use the diagram in p. 61 of the packet to help you.
1) Construct the midpoint quadrilateral of a *rectangle* below (same steps as in section #9 above).
To make it visually easier to follow, make your 4 circles have radius = AB.
2) Let the 4 midpoints be E, F, G, and H.
3) Connect the midpoints to form quadrilateral EFGH.
4) Draw the diagonals of the original rectangle (B to D, A to C).
5) These diagonals intersect the sides of EFGH in midpoints I, J, K, and L. Label these midpoints.
6) Connect these points to form quadrilateral IJKL.
7) Diagonals "EG" and "FH" intersect the sides of IJKL at their midpoints; connect these midpoints to form another quadrilateral.
8) In theory, you could keep doing this forever, creating smaller and smaller midpoint quadrilaterals.
Keep your circles visible, but lighter/thinner, as evidence of your work.
#10: The Midpoint Quadrilaterals of a Rectangle
Move the points of ABCD around. Observe how the midpoint quadrilaterals do - and don't - change. No matter how you manipulate rectangle ABCD, EFGH and IJKL should both be special quadrilaterals. Based on your work above... a) What special quadrilateral is always formed by the midpoints of a rectangle (i.e., what is EFGH)?
b) What special quadrilateral is always formed by the midpoints of THAT special quadrilateral (i.e. what is IJKL)?