Difference between revisions of "1961 AHSME Problems/Problem 8"
Rockmanex3 (talk | contribs) (Solution to Problem 8) |
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− | + | == Problem == | |
− | + | Let the two base angles of a triangle be <math>A</math> and <math>B</math>, with <math>B</math> larger than <math>A</math>. | |
+ | The altitude to the base divides the vertex angle <math>C</math> into two parts, <math>C_1</math> and <math>C_2</math>, with <math>C_2</math> adjacent to side <math>a</math>. Then: | ||
− | (B) | + | <math>\textbf{(A)}\ C_1+C_2=A+B \qquad |
+ | \textbf{(B)}\ C_1-C_2=B-A \qquad\\ | ||
+ | \textbf{(C)}\ C_1-C_2=A-B \qquad | ||
+ | \textbf{(D)}\ C_1+C_2=B-A\qquad | ||
+ | \textbf{(E)}\ C_1-C_2=A+B</math> | ||
− | + | ==Solution== | |
− | ( | + | <asy> |
+ | draw((0,0)--(120,0)--(40,50)--(0,0)); | ||
+ | draw((40,50)--(40,0)); | ||
+ | draw((40,4)--(44,4)--(44,0)); | ||
+ | label("$B$",(10,5)); | ||
+ | label("$A$",(100,5)); | ||
+ | label("$C_1$",(47,38)); | ||
+ | label("$C_2$",(34,33)); | ||
+ | </asy> | ||
+ | |||
+ | Noting that side <math>a</math> is opposite of angle <math>A</math>, label the diagram as shown. | ||
+ | |||
+ | From the two right triangles, | ||
+ | <cmath>B + C_2 = 90</cmath> | ||
+ | <cmath>A + C_1 = 90</cmath> | ||
+ | Substitute to get | ||
+ | <cmath>B + C_2 = A + C_1</cmath> | ||
+ | <cmath>B - A = C_1 - C_2</cmath> | ||
+ | The answer is <math>\boxed{\textbf{(B)}}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AHSME 40p box|year=1961|num-b=2|num-a=4}} | ||
− | |||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | |||
+ | [[Category:Introductory Geometry Problems]] |
Revision as of 14:13, 19 May 2018
Problem
Let the two base angles of a triangle be and , with larger than . The altitude to the base divides the vertex angle into two parts, and , with adjacent to side . Then:
Solution
Noting that side is opposite of angle , label the diagram as shown.
From the two right triangles, Substitute to get The answer is .
See Also
1961 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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All AHSME Problems and Solutions |
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