What's the area of the trapezoid? - July 8-12, 1996
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One diagonal of a square serves as the shorter base of a trapezoid, and a line through one of the vertices of the square contains the other base. The legs of the trapezoid are extensions of two sides of the square. If the area of the square is 2800, what is the area of the trapezoid?
This week there were lots of good explanations. We decided to show you answers from some fairly new people:Chia Wan Fen Grade: Grade 7 School: Raffles Girls' School (Secondary) This question uses the principle of similar triangles. The length of the trapezoid actually cuts the square into half. Half of the square, which is one triangle, is also 1/3 of the trapezoid. Area of one triangle = 2800/2 = 1400 Area of trapezoid = 1400X3 = 4200 The area of the trapezoid is 4200.
Chris Marsh Grade: between 11 and 12 School: Ukiah High School The answer could either be 4200 or 0. I think there are many ways to get the solution to this problem. The first thing I saw was that if the second base were directly on top of the first base (which makes it parallel to the first base and it goes go through a vertex of the square) the area would be zero. Realizing this was probably not the desired answer, I went on. Next I saw that if the second base were on a different vertex (one that the first base was not on) the two opposite sides of the square could be extended, forming a large triangle (of the 2 opposite sides and the second base). I saw that the triangle was made up of 4 smaller congruent triangles. The square was made of 2 of these smaller triangles and the trapezoid was made of 3 of these smaller triangles. Since I knew the area of the square was 2800 and the square was made up of two of these smaller triangles, each of the smaller triangles had to have an area of 1400. Since the trapezoid was made up of 3 of the smaller triangles its area had to be 3 * 1400 or 4200.
Lim Rongxuan Grade: 7 School: Raffles Girls' Secondary School Let us assume that the trapezoid is ABCD, AB being the shorter base and CD being the longer base. Let us call the vertex of the square touching line CD, E and the opposite vertex F. There are actually 4 equal triangles in the diagram, ABF, ABE, CBE and DAE. We know that ABF and ABE are equal as they are actually a square divided at its diagonal. Lines AB and ED are parallel because they are the sides of a trapezoid . Lines AD and BE must also be parallel as AD is the extension of FA and the sides of a square must be parallel. Thus, figure ABED must be be a parallogram. It is divided at its two vertices and the two triangles formed, which are ABE and DEA, must therefore be equal. The same reasoning will go to prove that triangles ABE and BEC are equal. Since the four triangles are equal and the area of two of them is 2800, the area of three of them (which is the area of the trapezoid) will be 2800 divided by 2 times 3 = 4200. The area of the trapezoid is then 4200.
Jason Yeung Grade: 9 School: Iolani School The area of the trapezoid is 4200. Since a trapezoid has 2 parallel bases, and since the sides of the trapezoid are the extensions of the sides of the square (like the transversal intersecting 2 parallel lines), the angles formed are corresponding angles and are both 45 degrees because the diagonal of the square makes a 45 degrees angle with the side. Since the leg of the trapezoid is the extension of the side of the square, a line is 180 degrees, and an angle of a square is 90 degrees, the remaining side of the triangle formed inside the trapezoid is 90 degrees. The remaining angle is 45 degrees, making the triangle a 45-45-90 isosceles triangle. The triangle is composed of a side of the square, extension of the square, and part of half the larger base of the triangle. Knowing that the area of the square is 2800, a side of the square would therefore be the sqrt of 2800, which also becomes the values of the sides of the isosceles triangle. The trapezoid is actually composed of 2 triangles like the above, and half the square. With the available fact, and knowing that the triangle formed inside the square by the diagnoal is also a 45-45-90 right triangle with legs of sqrt 2800, we proved that the three triangles in the trapezoid are congruent. area of a triangle is base * height / 2. sqrt of 2800 * sqrt of 2800 / 2 is 1400 area of 3 triangles is 1400 * 3, or 4200, the area of the trapezoid. Jason Yeung
Gregory W. Pack, Geometry teacher on summer vacation Pensacola Catholic High School Pensacola, FlThe formula for the area of a trapezoid is the product of one half the height and the sum of the bases. Height: AE is the height of trapezoid BDFG because diagonals of a square are perpendicular and the height of a trapezoid is the perpendicular length between bases. AE is one half AC since diagonals of a square bisect each other. The diagonal of a square is the length of a side times sqrt(2). The length of a side of a square is sqrt of its area. Calculating the height: The side of square ABCD = sqrt (2800) = 20*sqrt(7) Length of diagonal AC = AC = sqrt(2)*20*sqrt(7) = 20*sqrt(14) Length of height AE = AE = (1/2)20*sqrt(14) = 10*sqrt(14) Base one: BD = length of a diagonal = BD = 20*sqrt(14) Base two:Consider triangle ACG. Angle CAG is a right angle (angle AEB is a rightangle formed by perpendicular diagonals of a square and angle AEB issupplementary with interior angle CAG on the same side of transversal AE). The measure of angle ACG is 45 degrees (diagonals bisect opposite angles of a square which has all right angles by definition). The sum of the measures of any triangle is 180 degrees. Hence the measure of angle AGC is 45 degrees. With congruent base angles, triangle AGC is isosceles. Therefore AG = AC = 20*sqrt(14). Similarly, triangle ACF is also isosceles and FA = AC = 20*sqrt(14). Since A is between F and G, FG = FA + AG Base two: FG = 20*sqrt(14) + 20*sqrt(14) = 40*sqrt(14) Applying the trapezoid area formula: Area of trapezoid BDFG = [10*sqrt(14)/2][20*sqrt(14)+40*sqrt(14)] = [5*sqrt(14)][60*sqrt(14)] = (300*14) Area of trapezoid BDFG = 4200
Zeke - School Of Hard Knocks I used the infamous Gallagher Theorem, which specifies that a trapezoid of this nature will always have an area equivalent to 1.5 times that of the square.
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