GMAT Question of the Day - Data Sufficiency - Geometry/Triangles
If the length of side AC is equal to the length of side BC and the length of segment AD is equal to the length of segment BD is the length of segment CD less than half the value of the length of side AB?
1) The measure of angle ACB is not less than 90
2) The area of triangle ABC is 20
GMAT Question of the Day Solution
This question of the day is tricky but not tough if you stay organized. From the diagram and the given information we know that this is an isosceles triangle. Is it a right isosceles triangle? We don't know. However, if it is a right isosceles triangle then we can certainly calculate the values for all of the sides and determine the relationship between AB and CD. So one of the ways of answering this question comes down to determining whether triangle ABC is in fact a 45-45-90 triangle.
Statement 1) Not less than 90 means greater than or equal to 90. That means that we could have a right isosceles triangle. If we do have a right isosceles we can calculate the values for all of the sides based on the ratio 1:1:√2. That would make side CD exactly half of the value of AB. Meaning that we could answer NO to the question. Can we answer YES? Well, if the angle ABC is greater than 90 the hypotenuse will get longer while CD gets shorter. Hence CD will be less than half of AB (you can see this clearly in the diagram). So the answer would be YES. Both YES and NO means Insufficient.
2) It would have been great to have calculated the 45-45-90 scenario in advance. We shouldn't have assumed that this was a 45-45-90 but because it is an easy scenario to calculate it is worth doing. This is something that I always try to encourage in my GMAT tutoring students: to take simple steps to create more useful information.
This area proves that the triangle is in fact a right isosceles so we can determine the values for all of the sides. Sufficient.