Answer:
9 and 12
Step-by-step explanation:
Pythagorean Theorem. One of the most common Pythagorean triples is 3, 4, 5 with 3 and 4 the legs and 5 the hypoenuse. That will make 9, 12, 15 anotther trible because you are tripling the side lengths. 12-9=3 so this triangle works.
Part (a)
The domain is the set of allowed x inputs of a function.
The graph shows that x = 0 is not allowed because of the vertical asymptote located here. It seems like any other x value is fine though.
<h3>
Domain: set of all real numbers but 
</h3>
To write this in interval notation, we can say 
which is the result of poking a hole at 0 on the real number line.
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The range deals with the y values. The graph makes it seem like it stretches on forever in both up and down directions. If this is the case, then the range is the set of all real numbers.
<h3>
Range: Set of all real numbers</h3>
In interval notation, we would say 
which is almost identical to the interval notation of the domain, except this time of course we aren't poking at hole at 0.
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Part (b)
<h3>The x intercepts are x = -4 and x = 4</h3>
We can compact that to the notation 
These are the locations where the blue hyperbolic curve crosses the x axis.
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Part (c)
<h3>Answer: There aren't any horizontal asymptotes in this graph.</h3>
Reason: The presence of an oblique asymptote cancels out any potential for a horizontal asymptote.
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Part (d)
The vertical asymptote is located at x = 0, so the equation of the vertical asymptote is naturally x = 0. Every point on the vertical dashed line has an x coordinate of zero. The y coordinate can be anything you want.
<h3>Answer: x = 0 is the vertical asymptote</h3>
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Part (e)
The oblique or slant asymptote is the diagonal dashed line.
It goes through (0,0) and (2,6)
The equation of the line through those points is y = 3x
If you were to zoom out on the graph (if possible), then you should notice the branches of the hyperbola stretch forever upward but they slowly should approach the "fencing" that is y = 3x. The same goes for the vertical asymptote as well of course.
<h3>Answer: Oblique asymptote is y = 3x</h3>
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</span><span>Suppose you have a triangle with sides {6,7,8} — how do you find the height?This is a question some GMAT test takers ask. They know they would need the height to find the area, so they worry: how would I find that height. The short answer is:fuhgeddaboudit! Which height?First of all, the “height” of a triangle is its altitude. Any triangle has three altitudes, and therefore has three heights. You see, any side can be a base. From any one vertex, you can draw a line that is perpendicular to the opposite base — that’s the altitude to this base. Any triangle has three altitudes and three bases. You can use any one altitude-base pair to find the area of the triangle, via the formula A = (1/2)bh.
In each of those diagrams, the triangle ABC is the same. The green line is the altitude, the “height”, and the side with the red perpendicular square on it is the “base.” All three sides of the triangle get a turn. Finding a heightGiven the lengths of three sides of a triangle, the only way one would be able to find a height and the area from the sides alone would involve trigonometry, which is well beyond the scope of the GMAT. You are 100% NOT responsible for knowing how to perform these calculations. This is several levels of advanced stuff beyond the math you need to know. Don’t worry about that stuff.In practice, if the GMAT problem wants you to calculate the area of a triangle, they would have to give you the height. The only exception would be a right triangle — in a right triangle, if one of the legs is the base, the other leg is the altitude, the height, so it’s particularly easy to find the area of right triangles. Some “more than you need to know” caveatsIf you don’t want to know anything about this topic that you don’t absolutely need for the GMAT, skip this section!a. Technically, if you know the three sides of a triangle, you could find the area from something called Heron’s formula, but that’s also more than the GMAT will expect you to know. More than you needed to know!b. If one of the angles of the triangle is obtuse, then the altitudes to either base adjacent to this obtuse angle are outside of the triangle. Super-technically, an altitude is not a segment through a vertex perpendicular to the opposite base, but instead, a segment through a vertex perpendicular to the line containing the opposite base.In the diagram above, in triangle DEF, one of the three altitudes is DG, which goes from vertex D to the infinite straight line that contains side EF. That’s a technicality the GMAT will not test or expect you to know. Again, more than you needed to know!c. If the three sides of a triangle are all nice pretty positive integers, then in all likelihood, the actual mathematical value of the altitudes will be ugly decimals. Many GMAT prep sources and teachers in general will gloss over that, and for the purposes of easy problem-solving, give you a nice pretty positive integer for the altitude also. For example, the real value of the altitude from C to AB in the 6-7-8 triangle at the top is:Not only are you 100% NOT expected to know how to find that number, but also most GMAT practice question writers will spare you the ugly details and just tell you, for example, altitude = 5. That makes it very easy to calculate the area. Yes, technically, it’s a white lie, but one that spares the poor students a bunch of ugly decimal math with which they needn’t concern themselves. Actually, math teachers of all levels do this all the time — little white mathematical lies, to spare students details they don’t need to know.So far as I can tell, the folks who write the GMAT itself are sticklers for truth of all kinds, and do not even do this “simplify things for the student” kind of white lying. They are more likely to circumvent the entire issue, for example, by making all the relevant lengths variables or something like that. Yet again, more than you needed to know! What you need to knowYou need to know basic geometry. Yes, there is tons of math beyond this, and tons more you could know about triangles and their properties, but you are not responsible for any of that. You just need to know the basic geometry of triangles, including the formula A = (1/2)b*h. If the triangle is not a right triangle, you have absolute no responsibility for knowing how to find the height — it will always be given if you need it. Here’s a free practice question for you.</span>
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