Difference between revisions of "1996 AHSME Problems/Problem 21"
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| + | ==Problem== | ||
| + | |||
| + | Triangles <math>ABC</math> and <math>ABD</math> are isosceles with <math>AB=AC=BD</math>, and <math>BD</math> intersects <math>AC</math> at <math>E</math>. If <math>BD</math> is perpendicular to <math>AC</math>, then <math> \angle C+\angle D </math> is | ||
| + | |||
| + | <asy> | ||
| + | size(120); | ||
| + | pair B=origin, A=1*dir(70), M=foot(A, B, (3,0)), C=reflect(A, M)*B, E=foot(B, A, C), D=1*dir(20); | ||
| + | dot(A^^B^^C^^D^^E); | ||
| + | draw(A--D--B--A--C--B); | ||
| + | markscalefactor=0.005; | ||
| + | draw(rightanglemark(A, E, B)); | ||
| + | dot(A^^B^^C^^D^^E); | ||
| + | pair point=midpoint(A--M); | ||
| + | label("$A$", A, dir(point--A)); | ||
| + | label("$B$", B, dir(point--B)); | ||
| + | label("$C$", C, dir(point--C)); | ||
| + | label("$D$", D, dir(point--D)); | ||
| + | label("$E$", E, dir(point--E)); | ||
| + | </asy> | ||
| + | |||
| + | <math> \text{(A)}\ 115^\circ\qquad\text{(B)}\ 120^\circ\qquad\text{(C)}\ 130^\circ\qquad\text{(D)}\ 135^\circ\qquad\text{(E)}\ \text{not uniquely determined} </math> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | |||
==See also== | ==See also== | ||
{{AHSME box|year=1996|num-b=20|num-a=22}} | {{AHSME box|year=1996|num-b=20|num-a=22}} | ||
Revision as of 21:02, 19 August 2011
Problem
Triangles 
and 
are isosceles with 
, and 
intersects 
at 
. If is perpendicular to , then 
is
![[asy] size(120); pair B=origin, A=1*dir(70), M=foot(A, B, (3,0)), C=reflect(A, M)*B, E=foot(B, A, C), D=1*dir(20); dot(A^^B^^C^^D^^E); draw(A--D--B--A--C--B); markscalefactor=0.005; draw(rightanglemark(A, E, B)); dot(A^^B^^C^^D^^E); pair point=midpoint(A--M); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); [/asy]](http://latex.artofproblemsolving.com/c/7/4/c74dbdbc2ea44c18c583ca2b60609b48b4fbc1fd.png)
Solution
See also
| 1996 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
