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Since the triangle has ABC as a right triangle, the square of the triangle would be multiple of two non hypothenuse sides of the triangle divided by two. Statement ! just provides us with the information about the measurement of side AB Not sufficient.
Hence, we know that 2) The product of the non-hypotenuse sides is equal to 24, which is multiple of sides AB and BC (reuqired sides) We can conclude that Statement 2 is sufficient to answer the question.
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As per wikipedia, there are 3 ways to calculate the area of a right triangle:
"As with any triangle, to calculate the area, multiply the base and the corresponding height, and divide it by two. If ABC is a right triangle in A, each of the sides [AB] and [AC] can be considered as the height; the base is then the other side of the right angle ([AC] and [AB], respectively)."
Finally: "The area of the triangle could also be calculated by using the hypotenuse as the base. One would then have to calculate the height associated with the hypotenuse, as it would no longer be one of the sides."
Therefore, in the picture in wikipedia, besides using the legs, the only other way to draw a "height" of the triangle is to draw a line from the right angle vertex (A) to the hypotenuse (BC).
"If distinct points A, B, C, and D form a right triangle ABC with a height BD, what is the value of AB times BC?"
The language of this question appears at odds with the author's intent.
With respect to a triangle, the word "height" really only means "a line segment perpendicular to a triangle's edge, of a length equal to the distance of that edge from the opposite vertex." Height BD may or may not land D on the perimeter of the triangle. A line segment labeled "height" may just as easily be outside the triangle as inside the triangle, and is indeed outside the triangle in one example from a Grade 6 math text. "Height" means whatever helps you calculate the area of a triangle most easily; GMAC's OG12 does not use this term.
The term "altitude" would make the answer explanation work. From OG12 p.130: "The altitude of a triangle is the segment drawn from a vertex perpendicular to the side opposite that vertex." D must then lie on the perimeter of right triangle ABC, and B must be opposite the hypotenuse AC for D to be distinct from A and C.
Since GMAC does apparently regard "altitude" as fair game on the exam, and even gives the area of a triangle as "0.5*(base plus altitude)" on OG12 p.130, is there any chance of having this question amended to refer to "altitude BD" rather than "height BD" for future M04 takers?
Since the stimulus states the triangle ABC is a right triangle, D must lie on one of the sides.
also, One might assume the following, If we draw the right triangle ABC with D lieing on AC. S1 tells us that AB = 6. Since this is a right triangle, then the other sides are 8 and 10 by using the pythagorean triples 3-4-5. (use a factor of 2)
However, the sides could also be 2.5, 6, 6.5 using the pyth. triple of 5-12-13 (use a factor of 1/2)
Therefore S1 is not sufficient.
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If you draw the triangle you will see that AB and BC are the 2 short sides of the right triangle. You are asked for AB*BC, but stmt 1 doesn't say anything about BC. Stmt 2 instead gives you exactly the answer (you can read it as AB*BC=24). That's why the answer is B!