Chapter 6 Section 6: Trapezoids & Kites

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Chapter 6 Section 6: Trapezoids & Kites Goal: Recognize the properties of Kites & Trapezoids. Goal: Use the properties of Trapezoids & Kites to find missing measures. A KITE is a Quadrilateral that has TWO (2) pairs of Congruent-Consecutive Sides BUT Opposite Sides are NOT Congruent. The Diagonals of a KITE are Perpendicular—SEE Page 427 Theorem 6-6-1 IF a Quadrilateral is a KITE, EXACTLY one pair of Opposite Angles are Congruent AND the other Pair of Opposite Angles are Bisected by the Diagonal. SEE Page 427 Theorem 6-6-2 A TRAPEZOID is a Quadrilateral with EXACTLY 1 Pair of Parallel Sides. The Parallel sides are called the BASES and the NON-Parallel sides are called LEGS. A Trapezoid has two (2) sets of Base Angles. BASE ANGLES are formed where the Legs intersect the Bases. An ISOSCELES Trapezoid has exactly 1 pair of parallel sides and the legs are Congruent. A RIGHT Trapezoid has 1 pair of parallel sides and one (1) leg forms TWO 90o angles. Recall that in an Isosceles Triangle the BASE Angles are Congruent. When you “cut” the top off of an Isosceles Triangle an ISOSCELES TRAPEZOID is formed. The Two \ / Legs are congruent and the Base Angles have not changes and they remain Congruent. SEE PAGE 429—Theorem 6-6-3 and Theorem 6-6-4 THEOREM 6-6-5 “A Trapezoid is Isosceles IFF its Diagonals are Congruent.” BECAUSE: In an Isosceles Trapezoid, the legs are Congruent by Definition AND the Base Angles are also \ / Congruent Notice that there are TWO Triangles that can be shown to be Congruent by the SAS Postulate. || Once the Triangles are shown to be Congruent then EVERYTHING about them is Congruent INCLUDING the remaining Side which happens to be the Diagonals A MID-SEGMENT of a Trapezoid is very similar to the Mid-Segment of a Triangle in that it connects the Mid-Points of the two sides, SPECIFICALLY IN A TRAPEZOID, it connects the Mid-Points of the two Legs. TWO CHARACTERISTICS of the Mid-Segment: 1. It is Parallel to BOTH Bases 2. The LENGTH of the Mid-Segment is ½ the SUM of the length of BOTH Bases: MidSgmnt = ½ (b1 + b2) (SEE Page 431 Theorem 6-6-6) .

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Build a Tetrahedral Kite
Aeronautics Research Mission Directorate Build a Tetrahedral Kite Suggested Grades: 8-12 Activity Overview Time: 90-120 minutes In this activity, you will build a tetrahedral kite from Materials household supplies. • 24 straws (8 inches or less) - NOTE: The straws need to be Steps straight and the same length. If only flexible straws are available, 1. Cut a length of yarn/string 4 feet long. then cut off the flexible portion. • Two or three large spools of 2. Take six straws and place them on a flat surface. cotton string or yarn (approximately 100 feet total) 3. Use your piece of string to join three straws • Scissors together in a triangular shape. On the side where • Hot glue gun and hot glue sticks the two strings are extending from it, one end • Ruler or dowel rod for kite bridle should be approximately 20 inches long, and the • Four pieces of tissue paper (24 x other should be approximately 4 inches long. 18 inches or larger) See Figure 1. • All-purpose glue stick Figure 1 4. Tie these two ends of the string tightly together. Make sure there is no room for the triangle to wiggle. 5. The three straws should form a tight triangle. 6. Cut another 4-inch piece of string. 7. Take one end of the 4-inch string, and tie that end to a corner of the triangle that does not have the string ends extending from it. Figure 2. 8. Add two more straws onto the longest piece of string. 9. Next, take the string that holds the two additional straws and tie it to the end of one of the 4-inch strings to make another tight triangle.
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When Is a Tangential Quadrilateral a Kite?
Forum Geometricorum b Volume 11 (2011) 165–174. b b FORUM GEOM ISSN 1534-1178 When is a Tangential Quadrilateral a Kite? Martin Josefsson Abstract. We prove 13 necessary and sufﬁcient conditions for a tangential quadri- lateral to be a kite. 1. Introduction A tangential quadrilateral is a quadrilateral that has an incircle. A convex quadrilateral with the sides a, b, c, d is tangential if and only if a + c = b + d (1) according to the Pitot theorem [1, pp.65–67]. A kite is a quadrilateral that has two pairs of congruent adjacent sides. Thus all kites has an incircle since its sides satisfy (1). The question we will answer here concerns the converse, that is, what additional property a tangential quadrilateral must have to be a kite? We shall prove 13 such conditions. To prove two of them we will use a formula for the area of a tangential quadrilateral that is not so well known, so we prove it here ﬁrst. It is given as a problem in [4, p.29]. Theorem 1. A tangential quadrilateral with sides a, b, c, d and diagonals p,q has the area 1 2 2 K = 2 (pq) − (ac − bd) . p Proof. A convex quadrilateral with sides a, b, c, d and diagonals p,q has the area 1 2 2 2 2 2 2 2 K = 4 4p q − (a − b + c − d ) (2) p according to [6] and [14]. Squaring the Pitot theorem (1) yields 2 2 2 2 a + c + 2ac = b + d + 2bd. (3) Using this in (2), we get 1 2 2 K = 4 4(pq) − (2bd − 2ac) p and the formula follows.
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Cyclic and Bicentric Quadrilaterals G
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Volume 18 2018
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