M04-24

Question

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 .

Bunuel · Accepted Answer

Official Solution:

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. 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 = AC/2, from which we have that AC = 24. Sufficient.

Answer: B

tangt16 · Answer

How do we know that BD is median to AC and not AB or BC?

Bunuel · Answer

Draw triangle ABC. The median BD (the median from vertex B) can be only from B to side AC.

narmadadhruv · Answer

Hi, Towards the proof for the question, pls refer: https://www.algebra.com/algebra/homewor ... use.lesson rgds.

vb1991 · Answer

I think this is a  high-quality  question and I agree with the explanation.

A very clever manipulation of the 'triangle inside a semicircle' trope. Bravo!

abhola · Answer

Hi  Bunuel ,

How can we surely say that if a triangle follows AB^2 + BC^2 = AC^2, it is a rt. angles triangle? Let's say we have a 3,4,5 triangle which follows this property of AB^2 + BC^2 = AC^2 (AB or BC = 3 or 4 and AC = 5), now triangle ABC may be or may not be a rt. angles triangle.

But if a triangle is a rt. angles triangle, then only we can apply Pythagoras theorem.

Help me with this doubt here. Many Thanks!

Bunuel · Answer

The reverse of a Pythagorean theorem is also true: if the lengths of the sides of a triangle are a, b, and c, and a^2 + b^2 = c^2, then we have a right triangle.

Also, if the lengths of the sides of a triangle are a, b, and c, where the largest side is c, then:

For a right triangle:  a^2 +b^2= c^2 .
For an acute (a triangle that has all angles less than 90°) triangle:  a^2 +b^2>c^2 .
For an obtuse (a triangle that has an angle greater than 90°) triangle:  a^2 +b^2<c^2 .

Hoozan · Answer

Bunuel   avigutman   VeritasKarishma   IanStewart  is there any logical approach (instead of learning/knowing a theory) to this question?

IanStewart · Answer

You'd never see the word 'median' in a GMAT geometry question, so this is not presented the way an official question would be. A couple of the posts above explained how to solve the problem without relying on any theorems about triangle medians (theorems you'd never need to know for the test) -- essentially you just need to draw a circle around the triangle and use some GMAT-level facts about triangles inscribed in circles. From Statement 2:

- the triangle must be a right triangle if its sides obey Pythagoras
- if we draw a circle around a right triangle so the three corners of the triangle are on the circle, then the hypotenuse will always be a diameter of the circle
- the midpoint of the hypotenuse is therefore the center of the circle
- a 'median' of a triangle is a line connecting a corner of the triangle to the midpoint of the opposite side. The median in this question connects the midpoint of the hypotenuse to the opposite corner. Since the midpoint is the center of our circle and the opposite corner is on the circle, the median is a radius
- since we now know the radius is 12, the hypotenuse, which is a diameter, must be 24, and Statement 2 is sufficient

KarishmaB · Answer

You don't need to know all theorems involving medians, perpendicular bisectors and angle bisectors etc but in Geometry, it will help to know some basic properties. 
One of them is that if a triangle inscribed in a circle is right angled, its hypotenuse will be a diameter of the circle. This helps in obtaining the answer here (as shown by Ian above)

Abhishek008 · Answer

I think this is a high-quality question and I agree with the explanation.

A very clever manipulation of the 'triangle inside a semicircle' trope. Bravo!

Bunuel · Answer

The median of a triangle is a line from a vertex to the midpoint of the opposite side. In a right triangle the median drawn to the hypotenuse will coincide with the angle bisector in case the right triangle is also an isosceles one. In this case the median will also be an altitude (height) to the hypotenuse.

If a right triangle is NOT isosceles, then the median drawn to the hypotenuse will NOT cut the right angle in half (so it won't coincide with the angle bisector).

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

vedha0 · Answer

if one knows that median from right angle is half of the hypotenuse, then its a pretty easy question to answer in less than a minute! properties are key, sometimes!

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

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