Difference between revisions of "2014 AMC 10A Problems/Problem 16"
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~Lemma proof by sakshamsethi | ~Lemma proof by sakshamsethi | ||
+ | |||
+ | ==Solution 5 (Similarity)== | ||
+ | <asy> | ||
+ | import graph; | ||
+ | size(9cm); | ||
+ | pen dps = fontsize(10); defaultpen(dps); | ||
+ | pair D = (0,0); | ||
+ | pair F = (1/2,0); | ||
+ | pair C = (1,0); | ||
+ | pair G = (0,1); | ||
+ | pair E = (1,1); | ||
+ | pair A = (0,2); | ||
+ | pair B = (1,2); | ||
+ | pair H = (1/2,1); | ||
+ | |||
+ | // do not look | ||
+ | pair X = (1/3,2/3); | ||
+ | pair Y = (2/3,2/3); | ||
+ | |||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(G--E); | ||
+ | draw(A--F--B); | ||
+ | draw(D--H--C); | ||
+ | filldraw(H--X--F--Y--cycle,grey); | ||
+ | |||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,NE); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,SW); | ||
+ | label("$E$",E,E); | ||
+ | label("$F$",F,S); | ||
+ | label("$G$",G,W); | ||
+ | label("$H$",H,N); | ||
+ | |||
+ | label("$\frac12$",(0.25,0),S); | ||
+ | label("$\frac12$",(0.75,0),S); | ||
+ | label("$1$",(1,0.5),E); | ||
+ | label("$1$",(1,1.5),E); | ||
+ | </asy> | ||
+ | |||
+ | The area of the shaded area is the area of <math>\triangle DHC</math> minus the two triangles on the side. | ||
+ | Extend <math>\overline{DH}</math> so that it hits point <math>B</math>. Call the intersection of <math>\overline{AF}</math> and <math>\overline{DB}</math> point <math>P</math>. <cmath>\triangle APB \sim \triangle FPD</cmath> | ||
+ | Drop altitudes from <math>P</math> down to <math>\overline{DF}</math> and <math>\overline{AB}</math>; call the intersection points <math>L</math> and <math>M</math> respectively. | ||
+ | <cmath>\frac{DF}{AB}=\frac{LP}{PM}=\frac{\frac{1}{2}}{1}=\frac{1}{2}</cmath> | ||
+ | <cmath>LP=\frac{1}{3}\cdot LM=\frac{2}{3}</cmath> | ||
+ | Thus the two triangles on the side have area <math>\frac{1}{2} \cdot \frac{2}{2} \cdot \frac {1}{3} = \frac{1}{6}</math>. Since there are two, their total area is <math>2 \cdot \frac{1}{6} = \frac{1}{3}</math>. The area of <math>\triangle DHC</math> is <math>\frac{1}{2} \cdot 1 \cdot 1=\frac{1}{2}</math>. The shaded region is <math>\frac{1}{2}-\frac{1}{3} = \frac{1}{6}</math> which is <math>\boxed{\textbf{(E)} \frac{1}{6}}</math>. | ||
+ | |||
+ | ~JH. L | ||
==See Also== | ==See Also== |
Revision as of 03:04, 20 June 2022
Contents
Problem
In rectangle , , , and points , , and are midpoints of , , and , respectively. Point is the midpoint of . What is the area of the shaded region?
Solution 1
Denote . Then . Let the intersection of and be , and the intersection of and be . Then we want to find the coordinates of so we can find . From our points, the slope of is , and its -intercept is just . Thus the equation for is . We can also quickly find that the equation of is . Setting the equations equal, we have . Because of symmetry, we can see that the distance from to is also , so . Now the area of the kite is simply the product of the two diagonals over . Since the length , our answer is .
Solution 2
Let the area of the shaded region be . Let the other two vertices of the kite be and with closer to than . Note that . The area of is and the area of is . We will solve for the areas of and in terms of x by noting that the area of each triangle is the length of the perpendicular from to and to respectively. Because the area of = based on the area of a kite formula, for diagonals of length and , . So each perpendicular is length . So taking our numbers and plugging them into gives us Solving this equation for gives us
Solution 3
From the diagram in Solution 1, let be the height of and be the height of . It is clear that their sum is as they are parallel to . Let be the ratio of the sides of the similar triangles and , which are similar because is parallel to and the triangles share angle . Then , as 2 is the height of . Since and are similar for the same reasons as and , the height of will be equal to the base, like in , making . However, is also the base of , so where so . Subbing into gives a system of linear equations, and . Solving yields and , and since the area of the kite is simply the product of the two diagonals over and , our answer is .
Solution 4
Let the unmarked vertices of the shaded area be labeled and , with being closer to than . Noting that kite can be split into triangles and .
Lemma: The distance from line segment to is half the distance from to
Proof: Drop perpendiculars of triangles and to line , and let the point of intersection be . Note that and are similar to and , respectively. Now, the ratio of to is , which shows that the ratio of to is , because of similar triangles as described above. Similarly, the ratio of to is . Since these two triangles contain the same base, , the ratio of .
Because kite is orthodiagonal, we multiply
~Lemma proof by sakshamsethi
Solution 5 (Similarity)
The area of the shaded area is the area of minus the two triangles on the side. Extend so that it hits point . Call the intersection of and point . Drop altitudes from down to and ; call the intersection points and respectively. Thus the two triangles on the side have area . Since there are two, their total area is . The area of is . The shaded region is which is .
~JH. L
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.