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# In the figure above, D is a point on the side AC of ΔABC.Is ΔABC isos?

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Math Expert
Joined: 02 Sep 2009
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In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosc  [#permalink]

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09 Dec 2014, 15:39
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45% (medium)

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68% (01:34) correct 32% (01:33) wrong based on 109 sessions

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Tough and Tricky questions: Geometry.

In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosceles?

(1) The area of triangular region ABD is equal to the area of triangular region DBC.
(2) BD┴AC and AD = DC

Kudos for a correct solution.

Attachment:

triang ABCD.JPG [ 3.38 KiB | Viewed 3821 times ]

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Re: In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosc  [#permalink]

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09 Dec 2014, 20:53
1
Statement 1: The area of triangular region ABD is equal to the area of triangular region DBC.

This means that the point D is mid-point of AC. But, we can't still say whether the triangle ABC is isosceles or not. Insufficient.

Statement 2: BD┴AC and AD = DC. Signifies that the median drawn from the vertex B is a perpendicular bisector. This can only happen if the triangle is isosceles triangle with AB = BC. Sufficient

The answer should be B
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Re: In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosc  [#permalink]

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10 Dec 2014, 05:24
In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosceles?

(1) The area of triangular region ABD is equal to the area of triangular region DBC --> this gives us info that D is the midpoint of AC but we don't have any info about angle ADB & BDC....insufficient
(2) BD┴AC and AD = DC --> this gives us info that D is midpoint of AC and that BD┴AC, this condition will make AB=BC.......sufficient.(refer below image, hope my understanding is correct )

Ans. B)
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triangle.png [ 7.79 KiB | Viewed 3158 times ]

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Re: In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosc  [#permalink]

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26 Dec 2014, 11:01
Bunuel wrote:

Tough and Tricky questions: Geometry.

In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosceles?

(1) The area of triangular region ABD is equal to the area of triangular region DBC.
(2) BD┴AC and AD = DC

Kudos for a correct solution.

Attachment:
triang ABCD.JPG

The correct answer is B.
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Re: In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosc  [#permalink]

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25 Jul 2015, 02:43
What exactly does this "BD┴AC" say?

I don't know what the symbol means.
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Posts: 50002
Re: In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosc  [#permalink]

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25 Jul 2015, 02:44
noTh1ng wrote:
What exactly does this "BD┴AC" say?

I don't know what the symbol means.

It means that BD is perpendicular to AC.
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Re: In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosc  [#permalink]

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25 Jul 2015, 05:00
4
Bunuel wrote:

Tough and Tricky questions: Geometry.

In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosceles?

(1) The area of triangular region ABD is equal to the area of triangular region DBC.
(2) BD┴AC and AD = DC

Kudos for a correct solution.

Attachment:
The attachment triang ABCD.JPG is no longer available

Per statement 1, Area Triangle (BAD} = Area Triangle {BDC}

---> 0.5*BE*AD = 0.5*BE*CD ---> AD = CD . But no information about AB and BC. Thus this triangle may or may not be isosceles. Thus this statement is not sufficient.

Per statement 2, BD┴AC and AD = DC ---> Triangles BDA and BDC are congruent. ---> AB = BC ---> Triangle ABC is isosceles. Thus this statement is sufficient.

B is the correct answer.
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2015-07-25_7-55-54.jpg [ 20.54 KiB | Viewed 2757 times ]

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In the figure above, D is a point on the side AC of ΔABC.Is ΔABC isos?  [#permalink]

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20 Feb 2016, 09:42
Attachment:

1.png [ 16.33 KiB | Viewed 2356 times ]

In the figure above, D is a point on the side AC of ΔABC. Is ΔABC isosceles?

(1) The area of the triangular region ABD is equal to the area of triangular region DBC.

(2) BD is perpendicular to AC and AD = DC.
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In the figure above, D is a point on the side AC of ΔABC.Is ΔABC isos?  [#permalink]

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20 Feb 2016, 10:03
Nevernevergiveup wrote:
Attachment:
1.png

In the figure above, D is a point on the side AC of ΔABC. Is ΔABC isosceles?

(1) The area of the triangular region ABD is equal to the area of triangular region DBC.

(2) BD is perpendicular to AC and AD = DC.

Statement 1 is straightforward to see that it is NOT sufficient. The most this statement will do is to provide a scenario when we have AD=CD. But this does not yet answer the question asked.

Statement 2, is tricky as in from the figure provided, BD can either be to the left of A or between A and C or to the right of C. But in only 1 case (of between A and C) will you get AD=CD such that BD $$\perp$$ AC. With this scenario, triangle ABC is indeed isosceles (by making triangles ABD and BCD congruent to each other by the SAS property ---> AB=BC) . Hence sufficient.

B is the correct answer.

Hope this helps.
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Re: In the figure above, D is a point on the side AC of ΔABC.Is ΔABC isos?  [#permalink]

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20 Feb 2016, 18:46

We are after: Is ΔABC isosceles?

(1) Not sufficient. Using the formula for areas of triangle, we get

Area of ΔABD = Area of ΔABD
1/2 * AD * h = 1/2 * DC * h

h is the height for both triangles, if you drop a perpendicular from point B on AC. This will fall on the extended AC (if you can imagine it)

The above equation yields to:

But we want to know if ΔABC is isosceles. This information doesn't help so it is INSUFFICIENT.

(2) Sufficient

Since BD ⊥AC, and D is on AC, imagine D dividing AC into two sides i.e. AD =AC.

Now imagine two right angled Δs inside ABC being split in the middle by BD. These two right angled Δs have the same base and height so the hypotenuse will be same i.e. AB = AC
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Re: In the figure above, D is a point on the side AC of ΔABC.Is ΔABC isos?  [#permalink]

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17 Apr 2018, 20:44
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Re: In the figure above, D is a point on the side AC of ΔABC.Is ΔABC isos? &nbs [#permalink] 17 Apr 2018, 20:44
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# In the figure above, D is a point on the side AC of ΔABC.Is ΔABC isos?

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