Difference between revisions of "1977 AHSME Problems/Problem 9"
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+ | == Problem 9 == | ||
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//Credit to MSTang for the diagram | //Credit to MSTang for the diagram | ||
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+ | In the adjoining figure <math>\measuredangle E=40^\circ</math> and arc <math>AB</math>, arc <math>BC</math>, and arc <math>CD</math> all have equal length. Find the measure of <math>\measuredangle ACD</math>. | ||
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+ | <math>\textbf{(A) }10^\circ\qquad | ||
+ | \textbf{(B) }15^\circ\qquad | ||
+ | \textbf{(C) }20^\circ\qquad | ||
+ | \textbf{(D) }\left(\frac{45}{2}\right)^\circ\qquad | ||
+ | \textbf{(E) }30^\circ </math> | ||
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+ | |||
+ | ==Solution== | ||
+ | Solution by e_power_pi_times_i | ||
If arcs <math>AB</math>, <math>BC</math>, and <math>CD</math> are congruent, then <math>\measuredangle ACB = \measuredangle BDC = \measuredangle CBD = \theta</math>. Because <math>ABCD</math> is cyclic, <math>\measuredangle CAD = \measuredangle CBD = \theta</math>, and <math>\measuredangle ADB = \measuredangle ACB = \theta</math>. Then, <math>\measuredangle EAD = \measuredangle EDA = \dfrac{180^\circ - 40^\circ}{2} = 70^\circ</math>. <math>\theta = 55^\circ</math>. <math>\measuredangle ACD = 180^\circ - 55^\circ - 110^\circ = \boxed{\textbf{(B) }15^\circ}</math>. | If arcs <math>AB</math>, <math>BC</math>, and <math>CD</math> are congruent, then <math>\measuredangle ACB = \measuredangle BDC = \measuredangle CBD = \theta</math>. Because <math>ABCD</math> is cyclic, <math>\measuredangle CAD = \measuredangle CBD = \theta</math>, and <math>\measuredangle ADB = \measuredangle ACB = \theta</math>. Then, <math>\measuredangle EAD = \measuredangle EDA = \dfrac{180^\circ - 40^\circ}{2} = 70^\circ</math>. <math>\theta = 55^\circ</math>. <math>\measuredangle ACD = 180^\circ - 55^\circ - 110^\circ = \boxed{\textbf{(B) }15^\circ}</math>. |
Latest revision as of 12:30, 21 November 2016
Problem 9
In the adjoining figure and arc
, arc
, and arc
all have equal length. Find the measure of
.
Solution
Solution by e_power_pi_times_i
If arcs ,
, and
are congruent, then
. Because
is cyclic,
, and
. Then,
.
.
.