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# D is a point on the side AC of a triangle ABC. Is ABC

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D is a point on the side AC of a triangle ABC. Is ABC [#permalink]  17 Aug 2005, 11:05
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1. D is a point on the side AC of a triangle ABC. Is ABC isoceles?

1. The area of triangular region ABD = area of BDC.
2. BD perpendicular AC and AD = DC.

2. If the area of triangular region ABC is 25, what is the perimeter of ABC?

1. The length of one side of ABC is 5sqrt(2).
2. RST is a right isoceles triangle.
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Re: DS Triangles [#permalink]  17 Aug 2005, 11:24
Priti wrote:
1. D is a point on the side AC of a triangle ABC. Is ABC isoceles?

1. The area of triangular region ABD = area of BDC.
2. BD perpendicular AC and AD = DC.

2. If the area of triangular region ABC is 25, what is the perimeter of ABC?

1. The length of one side of ABC is 5sqrt(2).
2. RST is a right isoceles triangle.

B & B

Last edited by gotoknow3 on 18 Aug 2005, 04:36, edited 1 time in total.
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Gotoknow3, how did you get base = height for the 2 Q?
Is there any formula that gives the height, area of an isoceles triangle based on its equal sides?
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Re: DS Triangles [#permalink]  17 Aug 2005, 11:32
gotoknow3 wrote:
Priti wrote:
1. D is a point on the side AC of a triangle ABC. Is ABC isoceles?

1. The area of triangular region ABD = area of BDC.
2. BD perpendicular AC and AD = DC.

2. If the area of triangular region ABC is 25, what is the perimeter of ABC?

1. The length of one side of ABC is 5sqrt(2).
2. RST is a right isoceles triangle.

1. It can be equilateral or isoceles; E
2. st 1 gives base=height
st 2 gives information on the other side
so C

DO nto agree with inference from condition 1, It can be equilateral or isoceles, IMO there are lot more possiblities

Is equalateral trianlge is isoceles, since two sides are equal?
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Re: DS Triangles [#permalink]  17 Aug 2005, 12:43
Priti wrote:
1. D is a point on the side AC of a triangle ABC. Is ABC isoceles?

1. The area of triangular region ABD = area of BDC.
2. BD perpendicular AC and AD = DC.

S1 is insufficient.

S2 is sufficient. This is because Triangles BDC and BDA are both right triangles with BD as the height and DC = DA. 2 equal sides and included angle equal makes them congruent, therefore BA = BC.

Note that all equilateral triangles are isoscles as well. B is my choice.
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Quote:
2. If the area of triangular region ABC is 25, what is the perimeter of ABC?

1. The length of one side of ABC is 5sqrt(2).
2. RST is a right isoceles triangle.

I think S2 refers to ABC and not RST. Assuming that:

S1 is insufficient.

Consider S2.

Area of a right isoscles triangle is (a^2) /2 where a is the length of each of the equal sides. This is sufficient to calculate all 3 sides - 5 * sqrt(2), 5 * sqrt (2) and 10.

B is my choice.
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Re: DS Triangles [#permalink]  17 Aug 2005, 18:58
Priti wrote:
1. D is a point on the side AC of a triangle ABC. Is ABC isoceles?

1. The area of triangular region ABD = area of BDC.
2. BD perpendicular AC and AD = DC.

1. Insufficient.
Try draw a graph yourself and you'll know that this will be true if AD=DC, even if when ABC is not isoceles. If BD is perpendicular to AC then ABC is isoceles, otherwise it is not.

2. Sufficient
Triangle ABD and triangle CBD are similar (AD=CD, BD=BD, angle BDA=angle BDC), therefore BA=BC.

B

Quote:
2. If the area of triangular region ABC is 25, what is the perimeter of ABC?

1. The length of one side of ABC is 5sqrt(2).
2. ABC is a right isoceles triangle.

1. Insufficient.
2. Sufficient.
Say angle ABC is 90 degree, then we know that the other two angles are 45 degrees and the three sides are 1:1:sqrt(2). From S=1/2BA*BC=25 and BA=BC we can solve for BA, then the other two sides, and thus the perimeter.

B
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OAs are

1) B

2) B

Thanks everyone.
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Regarding First question, statement 2, the triangle could be equilateral, thus isosceles is not the only option.

Does isosceles mean exactly 2 equal sides or at least 2 equal sides?
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The isosceles is a special case of equilateral.
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