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Math Help - SAT Question

  1. #1
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    SAT Question

    What is the greatest possible area of a triangle with one side of length 7 and the another side of length 10?


    Choices:

    A - 17
    B - 34
    C - 35
    D - 70
    E - 140

    Answer is 35 but how?

    Thanks in advance
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  2. #2
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    Re: SAT Question

    Well, this is an ambiguous case, we must solve for the third side of the triangle. Frankly I forgot how to do this but I hope this site will help SparkNotes: Solving Oblique Triangles: The Ambiguous Case
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  3. #3
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    Re: SAT Question

    Thanks for the link. I thought about the law of sines as well but no trig is required for the SAT. I know there has to be an easier way. I am actually teaching the SAT course and cannot figure this one out.
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  4. #4
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    Re: SAT Question

    Quote Originally Posted by DaniNY View Post
    What is the greatest possible area of a triangle with one side of length 7 and the another side of length 10?
    Choices:
    A - 17
    B - 34
    C - 35
    D - 70
    E - 140
    Answer is 35 but how?
    If these are legs of a right triangle what is its area?
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  5. #5
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    Re: SAT Question

    If the lengths of two sides of a triangle are a and b and if the angle between them is \theta, then the area of the triangle is \frac{1}{2}ab sin\theta. You know the values of a and b so you are simply looking for the maximum value of sin\theta.
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  6. #6
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    Re: SAT Question

    Quote Originally Posted by Plato View Post
    If these are legs of a right triangle what is its area?

    It does not say whether it is a right triangle or not
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  7. #7
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    Re: SAT Question

    Quote Originally Posted by DaniNY View Post
    It does not say whether it is a right triangle or not
    a right triangle is going to have the maximum area of any triangle with given legs.
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  8. #8
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    Re: SAT Question

    Quote Originally Posted by DaniNY View Post
    It does not say whether it is a right triangle or not
    It does not have to.

    Suppose that the length of $\overline{AB}$ is 10. Think of a semicircle centered at $A$ with radius 7 beginning from the segment.

    If $C$ is any point on that semicircle not on $\overleftrightarrow {AB}$ then the max area for $\Delta ABC$ is when $\overline {AB} \bot \overline {AC} $
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  9. #9
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    Re: SAT Question

    ahhh yes! - thank you all for your help
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  10. #10
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    Re: SAT Question

    These tests dont want you to over think but everybody does, in its simplest form of 1/2 * base * height. .5 * 10*7 is 35. We all want to work out the third side as we should but as the angle Theta becomes Obtuse, what happens to the area as the angle widens and the third leg grows longer and the first two legs given spread wider apart, the base will stay the same and the height will grow shorter. Again you must assume that the leg that is growing or shrinking is not the base or height. A right triangle is the greatest Area with two fixed legs of length. Area of a triangle given three sides - Heron's Formula - Math Open Reference, something to play with.
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