Difference between revisions of "1985 AJHSME Problem 4"
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== Solution == | == Solution == | ||
Notice that the polygon is made up of two smaller rectangles with dimensions <math>5 \times 6</math> and <math>4 \times 4.</math> That makes the area <math>30 + 16 = 46,</math> so the answer is <math>\text{(C) 46.}</math> | Notice that the polygon is made up of two smaller rectangles with dimensions <math>5 \times 6</math> and <math>4 \times 4.</math> That makes the area <math>30 + 16 = 46,</math> so the answer is <math>\text{(C) 46.}</math> | ||
| + | |||
| + | == Easier Method == | ||
| + | Think of the polygon as a rectangle with a missing piece. We need to find the area of the rectangle, and the missing area. The polygon's length and width is 9 and 6, making the total area <math>9 \times 6</math> = 54. Now, we need to find the missing area. We see that the missing side's lengths are 2 and 4, making the area <math>4 \times 2</math> = 8. We got both the total area of the rectangle, and the missing area. <math>54 - 8</math> = 46, so the answer is <math>\text{(C) 46.}</math> | ||
Revision as of 14:43, 10 January 2025
Problem
The area of polygon 
, in square units, is
![[asy] draw((0,9)--(6,9)--(6,0)--(2,0)--(2,4)--(0,4)--cycle); label("A",(0,9),NW); label("B",(6,9),NE); label("C",(6,0),SE); label("D",(2,0),SW); label("E",(2,4),NE); label("F",(0,4),SW); label("6",(3,9),N); label("9",(6,4.5),E); label("4",(4,0),S); label("5",(0,6.5),W); [/asy]](http://latex.artofproblemsolving.com/a/4/1/a41b89f4f50edbafedb2363591428f6c617d5ebb.png)
Solution
Notice that the polygon is made up of two smaller rectangles with dimensions 
and 
That makes the area 
so the answer is 
Easier Method
Think of the polygon as a rectangle with a missing piece. We need to find the area of the rectangle, and the missing area. The polygon's length and width is 9 and 6, making the total area 
= 54. Now, we need to find the missing area. We see that the missing side's lengths are 2 and 4, making the area 
= 8. We got both the total area of the rectangle, and the missing area. 
= 46, so the answer is
