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19 Nov 2014, 16:28
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
38% (02:44) correct
62%
(02:54)
wrong
based on 194
sessions
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In the diagram below, if angle BAC is a right angle, is triangle ADC an equilateral triangle?
(1) ADC = 60 degrees
(2) EF is parallel to BC
Untitled.png [ 44.79 KiB | Viewed 2922 times ]
(1) ADC = 60 degrees
(2) EF is parallel to BC
Attachment:
Untitled.png [ 44.79 KiB | Viewed 2922 times ]
21 Nov 2014, 10:16
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Solution in the video below:
[youtube]https://www.youtube.com/watch?v=5tMBn-s_GGQ[/youtube]
[youtube]https://www.youtube.com/watch?v=5tMBn-s_GGQ[/youtube]
26 Jul 2020, 18:41
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To find: Whether triangle ADC is equilateral
To do this, we can either determine whether AD = DC = AC or that each interior angle = 60 degree
Statement 1: ADC = 60 degrees
At least one angle is equal to 60 degree, but we don't know whether AD = DC, so we can't say ADC is equilateral.
Insufficient (BCE)
Statement 2: EF is parallel to BC
This is a very interesting statement.
Let the point where EF and AD intersect be X
If EF is parallel to BC, then ADC = AXF (Corresponding angles)
Also, DFA = 90 and EAF = 90
Thus, EFA = b (since DAF has to be equal to EFA)
So, we have established that: in triangle AXF, XAF = XFA = b and AXF = ADC.
If we get to know the value of ADC, we will be able to determine whether triangle ADC is equilateral.
But, without concrete numbers, we can't say.
Insufficient (CE)
Let's combine the two statements now:
From statement 1 we get ADC = 60 degree
Thus, AXF = ADC = 60 degree
Applying angle sum property in triangle AXF, we have
2 b + AXF = 180
2b + 60 = 180
2b = 120
b = 60.
Now, in triangle ADC,
Angle ADC = 60 (acc to statement 1)
Angle DAC = b = 60 (just inferred by combining statements 1 and 2)
Thus, the third angle (ACD) will also be 60.
We can say for sure that triangle ADC is an equilateral triangel.
Sufficient
(C)
To do this, we can either determine whether AD = DC = AC or that each interior angle = 60 degree
Statement 1: ADC = 60 degrees
At least one angle is equal to 60 degree, but we don't know whether AD = DC, so we can't say ADC is equilateral.
Insufficient (BCE)
Statement 2: EF is parallel to BC
This is a very interesting statement.
Let the point where EF and AD intersect be X
If EF is parallel to BC, then ADC = AXF (Corresponding angles)
Also, DFA = 90 and EAF = 90
Thus, EFA = b (since DAF has to be equal to EFA)
So, we have established that: in triangle AXF, XAF = XFA = b and AXF = ADC.
If we get to know the value of ADC, we will be able to determine whether triangle ADC is equilateral.
But, without concrete numbers, we can't say.
Insufficient (CE)
Let's combine the two statements now:
From statement 1 we get ADC = 60 degree
Thus, AXF = ADC = 60 degree
Applying angle sum property in triangle AXF, we have
2 b + AXF = 180
2b + 60 = 180
2b = 120
b = 60.
Now, in triangle ADC,
Angle ADC = 60 (acc to statement 1)
Angle DAC = b = 60 (just inferred by combining statements 1 and 2)
Thus, the third angle (ACD) will also be 60.
We can say for sure that triangle ADC is an equilateral triangel.
Sufficient
(C)
