Difference between revisions of "2019 AMC 10A Problems/Problem 13"
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<cmath>\angle BEC+\angle EBC+\angle ECB=180^{\circ}\implies90^{\circ}+70^{\circ}+\angle ECB=180^{\circ}\implies\angle ECB=20^{\circ}</cmath> | <cmath>\angle BEC+\angle EBC+\angle ECB=180^{\circ}\implies90^{\circ}+70^{\circ}+\angle ECB=180^{\circ}\implies\angle ECB=20^{\circ}</cmath> | ||
− | Then, we take triangle <math>BFC</math>, and find <math>\angle BFC=180^{\circ}-50^{\circ}-20^{\circ}=\boxed{\textbf{(D) } 110}.</math> | + | Then, we take triangle <math>BFC</math>, and find <math>\angle BFC=180^{\circ}-50^{\circ}-20^{\circ}=\boxed{\textbf{(D) } 110^{\circ}}.</math> |
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==Solution 2== | ==Solution 2== | ||
Alternatively, we could have used similar triangles. We start similarly to Solution 1. | Alternatively, we could have used similar triangles. We start similarly to Solution 1. | ||
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Drawing it out, we see <math>\angle BDC</math> and <math>\angle BEC</math> are right angles, as they are inscribed in a semicircle. Therefore, <cmath>\angle BDA = 180^{\circ} - \angle BDC = 180^{\circ} - 90^{\circ} = 90^{\circ}.</cmath> | Drawing it out, we see <math>\angle BDC</math> and <math>\angle BEC</math> are right angles, as they are inscribed in a semicircle. Therefore, <cmath>\angle BDA = 180^{\circ} - \angle BDC = 180^{\circ} - 90^{\circ} = 90^{\circ}.</cmath> | ||
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So, <math>\triangle BEF \sim BDA</math> by AA Similarity, since <math>\angle EBF = \angle DBA</math> and <math>\angle BEC = 90^{\circ} = \angle BDA</math>. | So, <math>\triangle BEF \sim BDA</math> by AA Similarity, since <math>\angle EBF = \angle DBA</math> and <math>\angle BEC = 90^{\circ} = \angle BDA</math>. | ||
− | Thus, we know | + | Thus, we know <cmath>\angle EFB = \angle DAB = \angle CAB = 70^{\circ}.</cmath> |
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− | Finally, we | + | Finally, we deduce <cmath>\angle BFC = 180^{\circ} - \angle EFB = 180^{\circ} - 70^{\circ} = \boxed{\textbf{(D) } 110^{\circ}}.</cmath> |
− | + | ==Solution 3 (outside angles)== | |
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− | ==Solution 3== | ||
Through the property of angles formed by intersecting chords, we find that | Through the property of angles formed by intersecting chords, we find that | ||
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Adding the two equations gives us | Adding the two equations gives us | ||
− | <cmath>m\angle BFC | + | <cmath>m\angle BFC + m\angle CAB = m\overarc{BC}\implies m\angle BFC=m\overarc{BC} - m\angle CAB</cmath> |
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+ | Since <math>\overarc{BC}</math> is the diameter, <math>m\overarc{BC}=180^{\circ}</math>, and because <math>\triangle ABC</math> is isosceles and <math>m\angle ACB=40^{\circ}</math>, we have <math>m\angle CAB=70^{\circ}</math>. Thus | ||
+ | <cmath>m\angle BFC=180^{\circ}-70^{\circ}=\boxed{\textbf{(D) } 110^{\circ}}</cmath> | ||
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+ | ==Solution 4== | ||
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+ | Notice that if <math>\angle BEC = 90^{\circ}</math>, then <math>\angle BCE</math> and <math>\angle ACE</math> must be <math>20^{\circ}</math>. Using cyclic quadrilateral properties (or the properties of a subtended arc), we can find that <math>\angle EBD \cong \angle ECD = 20^{\circ}</math>. Thus <math>\angle CBF = 70 - 20 = 50^{\circ}</math>, and so <math>\angle BFC = 180 - 20 - 50 = 110^{\circ}</math>, which is <math>\boxed{\textbf{(D)}}</math>. | ||
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+ | ==Solution 5== | ||
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+ | <math>\triangle{ABC}</math> is isosceles so <math>\angle{CAB}=70^{\circ}</math>. Since <math>CB</math> is a diameter, <math>\angle{CDB}=\angle{CEB}=90^{\circ}</math>. Quadrilateral <math>ADFE</math> is cyclic since <math>\angle{ADF}+\angle{AEF}=180^{\circ}</math>. Therefore <math>\angle{BFC}=\angle{DFE}=180^{\circ}-\angle{CAB}=\boxed{110^{\circ}}</math> | ||
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+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/O_o_-yjGrOU?t=849 | ||
− | + | ~ pi_is_3.14 | |
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− | + | ==Video Solution== | |
− | + | https://youtu.be/GmQIEX4Izt4 | |
− | + | Education, the Study of Everything | |
==See Also== | ==See Also== |
Latest revision as of 11:15, 4 November 2022
Contents
Problem
Let be an isosceles triangle with
and
. Construct the circle with diameter
, and let
and
be the other intersection points of the circle with the sides
and
, respectively. Let
be the intersection of the diagonals of the quadrilateral
. What is the degree measure of
Solution 1
Drawing it out, we see and
are right angles, as they are inscribed in a semicircle. Using the fact that it is an isosceles triangle, we find
. We can find
and
by the triangle angle sum on
and
.
Then, we take triangle , and find
Solution 2
Alternatively, we could have used similar triangles. We start similarly to Solution 1.
Drawing it out, we see and
are right angles, as they are inscribed in a semicircle. Therefore,
So, by AA Similarity, since
and
.
Thus, we know
Finally, we deduce
Solution 3 (outside angles)
Through the property of angles formed by intersecting chords, we find that
Through the Outside Angles Theorem, we find that
Adding the two equations gives us
Since is the diameter,
, and because
is isosceles and
, we have
. Thus
Solution 4
Notice that if , then
and
must be
. Using cyclic quadrilateral properties (or the properties of a subtended arc), we can find that
. Thus
, and so
, which is
.
Solution 5
is isosceles so
. Since
is a diameter,
. Quadrilateral
is cyclic since
. Therefore
Video Solution by OmegaLearn
https://youtu.be/O_o_-yjGrOU?t=849
~ pi_is_3.14
Video Solution
Education, the Study of Everything
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.