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Math Expert V
Joined: 02 Sep 2009
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16 00:00

Difficulty:   85% (hard)

Question Stats: 35% (01:29) correct 65% (01:47) wrong based on 138 sessions

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The length of the median BD in triangle ABC is 12 centimeters, what is the length of side AC?

(1) ABC is an isosceles triangle.

(2) $$AC^2 = AB^2 + BC^2$$.

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Math Expert V
Joined: 02 Sep 2009
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2
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Official Solution:

(1) ABC is an isosceles triangle. Clearly insufficient.

(2) $$AC^2 = AB^2 + BC^2$$. This statement implies that ABC is a right triangle and AC is its hypotenuse. Important property: median from right angle is half of the hypotenuse, hence $$BD=12=\frac{AC}{2}$$, from which we have that $$AC=24$$. Sufficient.

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##### General Discussion
Intern  Joined: 14 Jul 2014
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could help me please what does it mean: "The length of the median BD" ... first I thought it is the midpoint of BC, but it is not ... and since english is not my mother tounge I do not know what it means
Math Expert V
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bigzoo wrote:
could help me please what does it mean: "The length of the median BD" ... first I thought it is the midpoint of BC, but it is not ... and since english is not my mother tounge I do not know what it means

The median of a triangle is a line from a vertex to the midpoint of the opposite side. Check for more HERE.
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Please correct my reasoning. I do not understand why A is not correct. Let's name each side of our equilateral triangle "x." AB=BC=AC=x. Since we are told BD is a median, it means AD=DC=x/2. In an equilateral triangle, a line drawn from one of the vertices to the midpoint of the opposite line will form a 90 degree angle. So we would have two equal triangles, AB-BD-DA and CB-BD-DC that each have one side, BD, length 12, one side, DC or DA, length x/2, and one side, BC or BA, length x. Since these are right triangles, (x/2)^2+12^2=x^2. We have an equation with one unknown. Why is this not sufficient? Bunuel can you help?
Math Expert V
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aaigla wrote:
Please correct my reasoning. I do not understand why A is not correct. Let's name each side of our equilateral triangle "x." AB=BC=AC=x. Since we are told BD is a median, it means AD=DC=x/2. In an equilateral triangle, a line drawn from one of the vertices to the midpoint of the opposite line will form a 90 degree angle. So we would have two equal triangles, AB-BD-DA and CB-BD-DC that each have one side, BD, length 12, one side, DC or DA, length x/2, and one side, BC or BA, length x. Since these are right triangles, (x/2)^2+12^2=x^2. We have an equation with one unknown. Why is this not sufficient? Bunuel can you help?

(1) says that ABC is isosceles, not equilateral.
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Hi Bunuel,

Could you please elaborate (as in how does it come about) on the geometrical property mentioned for the sufficiency of Statement 2 : "Important property: median from right angle is half of the hypotenuse"?

It would just make it easier to remember.

Thanks. Re: M28-40   [#permalink] 28 May 2020, 23:43

# M28-40

Moderators: chetan2u, Bunuel   