Difference between revisions of "1978 AHSME Problems/Problem 29"
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==Problem== | ==Problem== | ||
Sides <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math>, respectively of convex quadrilateral <math>ABCD</math> are extended past <math>B</math>, <math>C</math>, <math>D</math>, and <math>A</math> to points <math>B'</math>, <math>C'</math>, <math>D'</math>, and <math>A'</math>. Also, <math>AB = BB' = 6</math>, <math>BC = CC' = 7</math>, <math>CD = DD' = 8</math>, and <math>DA = AA' = 9</math>, and the area of <math>ABCD</math> is 10. Find the area of <math>A'B'C'D'</math>. | Sides <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math>, respectively of convex quadrilateral <math>ABCD</math> are extended past <math>B</math>, <math>C</math>, <math>D</math>, and <math>A</math> to points <math>B'</math>, <math>C'</math>, <math>D'</math>, and <math>A'</math>. Also, <math>AB = BB' = 6</math>, <math>BC = CC' = 7</math>, <math>CD = DD' = 8</math>, and <math>DA = AA' = 9</math>, and the area of <math>ABCD</math> is 10. Find the area of <math>A'B'C'D'</math>. | ||
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==Solution== | ==Solution== |
Revision as of 13:17, 23 August 2022
Problem
Sides , , , and , respectively of convex quadrilateral are extended past , , , and to points , , , and . Also, , , , and , and the area of is 10. Find the area of .
Solution
Notice that the area of is the same as that of (same base, same height). Thus, the area of is twice that (same height, twice the base). Similarly, [ ] = 2 [ ], and so on.
Adding all of these, we see that the area the four triangles around is twice [ ] + [ ] + [ ] + [ ], which is itself twice the area of the quadrilateral . Finally, [] = [] + 4 [] = 5 [] = .
~ Mathavi
Note: Anyone with a diagram would be of great help (still new to LaTex).