Difference between revisions of "1960 AHSME Problems/Problem 37"
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Since <math>CD</math> is perpendicular to <math>AB</math>, <math>ND = WX</math>. That means <math>CN = h-x</math>. | Since <math>CD</math> is perpendicular to <math>AB</math>, <math>ND = WX</math>. That means <math>CN = h-x</math>. | ||
The sides of the rectangle are parallel, so <math>XY \parallel WZ</math>. That means by AA Similarity, <math>\triangle CXY \sim \triangle CAB</math>. Letting <math>n</math> be the length of the base of the rectangle, that means | The sides of the rectangle are parallel, so <math>XY \parallel WZ</math>. That means by AA Similarity, <math>\triangle CXY \sim \triangle CAB</math>. Letting <math>n</math> be the length of the base of the rectangle, that means | ||
| − | <cmath>\frac{h-x}{n} = \frac{}{}</cmath> | + | <cmath>\frac{h-x}{n} = \frac{h}{b}</cmath> |
| + | <cmath>n = \frac{b(h-x)}{h}</cmath> | ||
| + | That means the area of the rectangle is | ||
==See Also== | ==See Also== | ||
{{AHSME 40p box|year=1960|num-b=36|num-a=38}} | {{AHSME 40p box|year=1960|num-b=36|num-a=38}} | ||
Revision as of 18:16, 15 May 2018
Problem
The base of a triangle is of length 
, and the altitude is of length 
.
A rectangle of height 
is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
Solution (WIP)
Let 
, 
, and 
.
![[asy] pair A=(0,0),B=(56,0),C=(20,48),D=(20,0),W=(10,0),X=(10,24),Y=(38,24),Z=(38,0); draw(A--B--C--A); draw((10,0)--(10,24)--(38,24)--(38,0)); draw(C--D); dot(A); dot(B); dot(C); dot(D); dot(W); dot(X); dot(Y); dot(Z); dot((20,24)); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,S); label("$W$",W,S); label("$X$",X,NW); label("$Y$",Y,NE); label("$Z$",Z,S); label("$N$",(20,24),NW); [/asy]](http://latex.artofproblemsolving.com/e/6/d/e6d29621bbd6dd27a480914b24417fe5401c4700.png)
Since 
is perpendicular to 
, 
. That means 
.
The sides of the rectangle are parallel, so 
. That means by AA Similarity, 
. Letting 
be the length of the base of the rectangle, that means
![\[\frac{h-x}{n} = \frac{h}{b}\]](http://latex.artofproblemsolving.com/c/9/7/c9722c0ba60eb5a15a89847af635926431c82a5f.png)
![\[n = \frac{b(h-x)}{h}\]](http://latex.artofproblemsolving.com/4/9/8/4980bcee0fe5b4ea52da94054984a33b34d8ba40.png)
That means the area of the rectangle is
See Also
| 1960 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 36 |
Followed by Problem 38 | |
| 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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
