In this packet, you will learn various methods for constructing special quadrilaterals, and you will explore various properties of special quadrilaterals.

[size=150][size=200][b][u]#1[/u]: Construct an Isosceles Triangle and a Kite[br][/b][/size][/size][b][i]Use the instructions and example on the front page of the packet to guide your construction.[/i][/b][br][br]1) Using the tools & strategies you have been using on previous assignments (circles, compass, etc.), build [b]kite KITE[/b] so that [br][list][*][color=#0000ff]MN = KI = TI [/color][/*][*][color=#0000ff]KE = TE[/color][/*][*][color=#0000ff]KE > KI[/color][/*][/list][br]2) [b]Label [/b]the points K, I, T, and E.[br][br]3) Draw the [b]diagonals [/b]of the kite as well.[br][br][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color]

[size=150][size=200][b][u]#2[/u]: Two Special Kites: A Rhombus and a Concave Kite[br][/b][/size][/size][b][i][br]Use the instructions and example on p. 11 in the packet to guide your construction.[/i][/b][br][br]If you make a kite with [b]four congruent sides[/b], you will end up making a [b]rhombus[/b].[br]Construct rhombus RHOM by doing the following:[br][br]1) Construct an isosceles triangle above "RO" (2 congruent circles, intersection, etc.).[br][br]2) Construct another isosceles triangle - this time with the same lengths - below "RO."[br][br]3) Label the top intersection "H" and the bottom intersection "M." RHOM is a rhombus, because each side is a radius of congruent circles.[br][br]4) Draw the other diagonal, "HM."[br][br][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color]

1. Using any radius, construct an isosceles triangle above the given base.[br][br]2. With a different radius, construct another isosceles triangle - also above the given base.[br][br]3. Label the new corner points I and E.[br][br]4. Draw the diagonal between I and E.[br][br][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color]

[u][b][size=150]Key properties to notice:[/size][/b][/u][br][br]1. The main diagonal of a kite is the [b]perpendicular bisector[/b] of the other diagonal! [br][i](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].)[/i][br][br]2. This means that the diagonals of a kite are always [b]perpendicular[/b], and[br][br]3. The main diagonal [b]bisects [/b]the other diagonal.[br][br]4. The main diagonal is the [b]angle bisector[/b] of each angle [i](this is prove-able with triangle congruence + CPCTC).[/i]

[size=150][size=200][b][u]#4[/u]: Find a Segment's Midpoint Using the Kite Construction[br][/b][/size][/size][b][i][br]Use the instructions and example on p. 14 of the packet to guide your construction.[/i][/b][br][br]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 [b]same radius[/b], centered at B.)[br][br]2) Label the intersection point above as [b]"C,"[/b] but [b]don't [/b]add the segments connecting it to A or B.[br][br]3) Repeat these steps below the segment, but with a [b][i]different [/i][/b]radius. [br][br]4) Label the intersection point below the segment as [b][color=#0000ff]"D."[/color][/b][br][br]5) Make a [color=#0000ff][b]line [/b][/color]that goes through C and D. This is the [color=#0000ff]perpendicular bisector[/color] of the original segment (prove-able through triangle congruence + CPCTC!).[br][br]6) Place a point where the lines intersect and label it [b][color=#0000ff]"M."[/color][/b] This is the [b][color=#0000ff]midpoint [/color][/b]of the segment. [br][br][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color]

[size=150][size=200][b][u]#5[/u]: Construct a Square from its Diagonal[br][/b][/size][/size][b][i][br]Use the instructions and example on p. 15 of the packet to guide your construction.[/i][/b][br][br]If you construct a rhombus, BUT you make it equiangular, you will produce a [b]square[/b].[br][br]Conveniently for us,[b] the four corners of a square lie on a circle[/b] (see diagram on p. 15).[br][br]Using "SU" below, construct the original square that has "SU" as its [i]diagonal[/i].[br][br]1) Construct the perpendicular bisector of "SU" (follow the midpoint steps from section #4 above).[br][br]2) Label midpoint of "SU", "R." [br][br]3) Make a circle with center R that goes through S.[br][br]4) Label where the circle crosses the perpendicular bisector, "Q" and "A."[br][br]5) Add segments "SQ," "UQ," "SA," and "UA." SQUA should now be a square![br][br][b][u]Check your work [/u][/b]by moving around the original segment, "SU." The square should adjust accordingly while remaining a square.[br][br][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color]

[size=150][size=200][b][u]#6[/u]: Bisect an Angle[br][/b][/size][/size][b][i][br]Use the instructions and example on p. 16 of the packet to guide your construction.[/i][/b][br][br]This is a review from the Chapter 6 assignment. Remember, to bisect an angle with construction tools, you can:[br][br]1) Construct a circle centered at B. Label where it intersects the two sides of the angle as [color=#0000ff][b]"A" [/b][/color]and [b][color=#0000ff]"C"[/color][/b].[br][br]2) Construct a circle centered at A.[br][br]3) With the COMPASS tool, use the same radius as circle A to construct a circle centered at C. [br][br]4) Label the intersection of circle A & circle C (the intersection that is further away from B) as [color=#0000ff][b]point D[/b][/color].[br][br]5) Construct a [b][color=#0000ff]ray [/color][/b]from B through D. This is the [color=#0000ff][b]angle bisector [/b][/color]of angle ABC. [br][br][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color]

[b][color=#9900ff][u]Check your work[/u]:[/color] [/b]use the [color=#9900ff][b]purple points[/b][/color] to click and [color=#9900ff][b]move around[/b] [/color]the angle. [br]The ray should always bisect the angle, no matter how wide or narrow you make the angle.[color=#0000ff][br][br][br]NOTICE [/color][color=#0000ff]that if you connected the points, [b]BADC[/b] would be a [u][b]kite[/b][/u]! [br][br][br][/color]Now, use the same procedure from #6a to bisect right angle P below.

[size=150][size=200][b][u]#7[/u]: Construct a Regular Octagon[br][/b][/size][/size][b][i][br][color=#ff0000]Use the instructions and example on p. 17 of the packet to guide your construction.[br][/color][/i][/b][color=#ff0000][br][b]^^^^ DEFINITELY make sure you're looking at the diagram on this page as you go.[/b][/color][br][br]A [b]regular octagon[/b] has eight congruent sides and eight congruent angles. Like a square, all 8 vertices of a regular octagon [b]lie on a circle[/b].[br][br]Follow the steps to construct a regular octagon below:[br][br]1) Construct the[color=#0000ff]perpendicular bisector of "AE."[/color][br][br]2) Label the midpoint of "AE,"[b][color=#0000ff] "M."[/color][/b][br][br]3) Place a [b][color=#0000ff]circle [/color][/b]with center M, passing through A.[br][br]4) Label the points where the circle intersects the perpendicular bisector: [b][color=#0000ff]"C"[/color][/b] and [color=#0000ff][b]"G." [/b][/color][br][br]NOTICE: M is like the "origin" where four right angles all meet. In the next steps, you will construct the [b]angle bisector[/b] of each of these right angles.[br][br]5) Place a [color=#0000ff]circle [/color]centered at A, through M. [br][br]6) Place a [color=#0000ff]circle [/color]centered at C, through M. [br][br]7) Place a [color=#0000ff]circle [/color]centered at E, through M. [br][br]8) Place a [color=#0000ff]circle [/color]centered at G, through M. [br][br]9) There should now be[b][color=#0000ff] four intersection points[/color][/b], one in each "quadrant" (top left, top right, bottom left, bottom right). For each intersection, draw a [b][color=#0000ff]ray [/color][/b]going through it, starting from M. These rays should now [color=#0000ff]bisect [/color]the original four right angles.[br][br]10) Label where the circle with center M intersects these rays - [b][color=#0000ff]"B" "D" "F" and "H"[/color][/b] - so that all the points are in [color=#0000ff]alphabetical order [/color]around the circle.[br][br]11) Add [b][color=#0000ff]segments [/color][/b]connecting the eight points on the circle to form a regular octagon ABCDEFGH![br][br][color=#ff0000][b][i]Keep your circles visible, but lighter/thinner, as evidence of your work.[/i][/b][/color]

[size=150][size=200][b][u]#8[/u]: Construct a Regular Dodecagon (Clock)[br][/b][/size][/size][br]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.[br][br]Can you see how the arcs you created formed four petals of a flower? [br]Can you see how to draw those petals using just four circles? [br]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. [br][br]1) On the segment below, [b]recreate[/b] the construction of the [b]four "petals" [/b]from the octagon, using the perpendicular bisector, circle M, and the four "outer" circles. Omit the angle bisectors, and omit the octagon. [br][br]2) Locate the 12 equidistant points on the circle, and use them to construct a [b]regular dodecagon[/b] - an equilateral and equiangular 12-gon. [br][br]3) Using the text-box feature, decorate your dodecagon with the numbers of a [b][color=#0000ff]clock[/color][/b]! [br][br][b][i][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color][/i][/b]

[size=150][size=200][b][u]#9[/u]: Join the Midpoints of Any Quadrilateral[br][/b][/size][/size][br][b][i][color=#ff0000]Use the diagram in p. 59 of the packet to help you. [/color][/i][/b][br][br]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.[br][br]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![br][br]2) Make a [b]circle [/b]centered at A with a[b] BIG radius[/b] (more than half the length of the longest side).[br][br]3) Use the COMPASS tool to make a circle with the[b] same radius[/b], centered at[b] B[/b].[br][br]4) Use the INTERSECT tool to place points at the two intersection points, then draw the segment connecting these two points.[br][br]5) Place a point[b][color=#0000ff] "M"[/color][/b] where the segment intersects side "AB." This is the midpoint![br][br]6) Repeat steps 3-5 using a circle with the same radius centered at [b]C[/b], to identify and label [b][color=#0000ff]"N,"[/color][/b] the midpoint of "BC."[br][br]7) Repeat steps 3-5 using a circle with the same radius centered at [b]D[/b], to identify and label [b][color=#0000ff]"O""[/color][/b] the midpoint of "CD."[br][br]8) Repeat steps 4-5 to identify and label [b][color=#0000ff]"P,"[/color][/b] the midpoint of "AD."[br][br]9) Connect the midpoints to form quadrilateral [b][color=#0000ff]MNOP[/color][/b].[br][br][b][i][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color][/i][/b]

[size=150][size=200][b][u]#10[/u]: Join the Midpoints of a Rectangle and Peer Into the Infinitesimal![br][/b][/size][/size][br][b][i][color=#ff0000]Use the diagram in p. 61 of the packet to help you. [/color][/i][/b][br][br]1) Construct the midpoint quadrilateral of a *rectangle* below (same steps as in section #9 above). [br]To make it visually easier to follow,[b] make your 4 circles have [/b][b][color=#0000ff]radius = AB[/color][/b]. [br]2) Let the [b]4 midpoints be[color=#0000ff] E, F, G, and H[/color].[/b][br][br]3) Connect the midpoints to form [b][color=#0000ff]quadrilateral EFGH[/color][/b]. [br][br]4) Draw the diagonals of the original rectangle (B to D, A to C).[br][br]5) These diagonals intersect the sides of EFGH in midpoints[color=#0000ff][b] I, J, K, and L[/b][/color]. Label these midpoints. [br][br]6) Connect these points to form [color=#0000ff][b]quadrilateral IJKL[/b][/color]. [br][br]7) Diagonals "EG" and "FH" intersect the sides of IJKL at their midpoints; connect [i]these [/i]midpoints to form [i]another [/i]quadrilateral.[br][br]8) In theory, you could keep doing this forever, creating smaller and smaller midpoint quadrilaterals. [br][br][b][i][color=#ff0000]Keep your circles visible, but lighter/thinner, as evidence of your work.[/color][/i][/b]