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# In ∆ ABC above, AD = DC. Which of the following must be true?

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Math Expert
Joined: 02 Sep 2009
Posts: 54370
In ∆ ABC above, AD = DC. Which of the following must be true?  [#permalink]

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09 Aug 2017, 11:10
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Difficulty:

45% (medium)

Question Stats:

62% (01:35) correct 38% (01:44) wrong based on 60 sessions

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In ∆ ABC above, AD = DC. Which of the following must be true?

(A) AB = BC
(B) x = y = 90
(C) Area of ∆ ABD = area of ∆ CBD
(D) Perimeter of ∆ ABD = perimeter of ∆ CBD
(E) Measure of angle BAD = measure of angle BCD

Attachment:

2017-08-09_2208.png [ 11.94 KiB | Viewed 904 times ]

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Re: In ∆ ABC above, AD = DC. Which of the following must be true?  [#permalink]

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09 Aug 2017, 11:57
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Bunuel wrote:

In ∆ ABC above, AD = DC. Which of the following must be true?

(A) AB = BC
(B) x = y = 90
(C) Area of ∆ ABD = area of ∆ CBD
(D) Perimeter of ∆ ABD = perimeter of ∆ CBD
(E) Measure of angle BAD = measure of angle BCD

Attachment:
2017-08-09_2208.png

if we drop a perpendicular from B on the base AC, then it will be the height for both ∆ ABD & ∆ CBD.
Now base of both these ∆s AD & DC are equal.
Hence Area of ∆ ABD = area of ∆ CBD
Option C
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Re: In ∆ ABC above, AD = DC. Which of the following must be true?  [#permalink]

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09 Aug 2017, 20:53
Bunuel wrote:

In ∆ ABC above, AD = DC. Which of the following must be true?

(A) AB = BC
(B) x = y = 90
(C) Area of ∆ ABD = area of ∆ CBD
(D) Perimeter of ∆ ABD = perimeter of ∆ CBD
(E) Measure of angle BAD = measure of angle BCD

Attachment:
2017-08-09_2208.png

A, E and D out as they are same options
B out as it is a bisector and not necessarily a perpendicular bisector (x and y can take other value as well apart from 90)

By POE C

i think height will be common for both the ∆ ABD & ∆ CBD while their bases are equal.
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Re: In ∆ ABC above, AD = DC. Which of the following must be true?  [#permalink]

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12 Aug 2017, 12:41
Option C

height of perpendicular form B to base AC is same for both triangles and value of base is also same since AD=DC
therefore it is clear that .....Area of ∆ ABD = area of ∆ CBD
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Re: In ∆ ABC above, AD = DC. Which of the following must be true?   [#permalink] 12 Aug 2017, 12:41
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