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04 Sep 2022, 10:53
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C
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Difficulty:
85%
(hard)
Question Stats:
44% (02:42) correct
56%
(03:24)
wrong
based on 36
sessions
History
Date
Time
Result
Not Attempted Yet
Note: Figure not drawn to scale.
In an isosceles triangle ABC shown above, AB = BC, and AC = CD = DF = FE = EB. What is the value of x ?
A. 10°
B. 15°
C. 20°
D. 30°
E. 40°
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Attachment:
Geometry.png
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04 Sep 2022, 12:29
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\(\triangle EBF\)
EF = EB
Therefore , \(\angle EBF = \angle BFE = x\)
\(\angle FEB = 180 - 2x \)
\(\triangle EFD\)
\(\angle DEF= 180- (180 - 2x) = 2x \)
EF = FD
Therefore , \(\angle DEF = \angle FDE = 2x\)
\(\angle EFD= 180- 4x\)
\(\triangle FDC\)
\(\angle DFC= 180- (180 - 3x) = 3x \)
DC = FD
Therefore , \(\angle DFC = \angle FCD = 3x\)
\(\angle FDC= 180- 6x \)
\(\triangle ADC\)
\(\angle ADC= 180- (180 - 4x) = 4x \)
DC = AC
Therefore , \(\angle ADC = \angle CDA = 4x\)
\(\triangle ABC\)
AB = BC
Therefore , \(\angle BAC = \angle BCA = 4x\)
Also, \(\angle BAC + \angle BCA + \angle ABC\) = 180
4x + 4x + x = 180
9x = 180
x = 20
IMO C
EF = EB
Therefore , \(\angle EBF = \angle BFE = x\)
\(\angle FEB = 180 - 2x \)
\(\triangle EFD\)
\(\angle DEF= 180- (180 - 2x) = 2x \)
EF = FD
Therefore , \(\angle DEF = \angle FDE = 2x\)
\(\angle EFD= 180- 4x\)
\(\triangle FDC\)
\(\angle DFC= 180- (180 - 3x) = 3x \)
DC = FD
Therefore , \(\angle DFC = \angle FCD = 3x\)
\(\angle FDC= 180- 6x \)
\(\triangle ADC\)
\(\angle ADC= 180- (180 - 4x) = 4x \)
DC = AC
Therefore , \(\angle ADC = \angle CDA = 4x\)
\(\triangle ABC\)
AB = BC
Therefore , \(\angle BAC = \angle BCA = 4x\)
Also, \(\angle BAC + \angle BCA + \angle ABC\) = 180
4x + 4x + x = 180
9x = 180
x = 20
IMO C
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11 Dec 2024, 10:17
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