Algebra

Find the coordinates of the midpoint of the hypotenuse of the right triangle whose vertices are A (1,1) B (5,2) C (4,6) and show that it is equidistant of each of the vertices.....
And can you please tell me how to prove it in drawing....

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  1. Your diagram should show that you have a right angle at B , but let's prove it anyway.

    slope AB = (2-1)/(5-1) = 1/4
    slope of BC = (6-2)/(4-5) = -4
    YES, AB is perpendicular to BC, thus AC is a hypotenuse

    midpoint of AC = M( (1+4)/2 , (1+6)/2) = M(5/2 , 7/2)
    = M(2.5 , 3/5)
    AM = √(1.5^2 + 2.5^2 = √8.5
    CM = √(1.5^2 + 2.5^2) = √8.5 , clearly has to be, since we found the midpoint
    BM = √((5 - 2.5)^2 + (2-3.5)^2 ) = √8.5

    YUP, all is good

    If you make a good sketch, you should be able to show it by counting the displacements along the grid pattern

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    posted by Reiny
  2. What is the solution when the formula used is distance between points??

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  3. I don't understand your question.
    I DID use the distance between two points formula, I just did not show the subtraction since that is easily done in your head.

    here is one of them with all steps :
    AM = √( 2.5 - 1)^2 + (3.5-1)^2 )
    = √( 1.5^2 + 2.5^2)
    = √2.25 + 6.25)
    = √8.5 ---- see above

    = appr 2.9155

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    posted by Reiny
  4. I mean is to get the distance of A B C you used distance formula not slope...

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  5. I think the main purpose of the question, even though they did not say that, was to show that the midpoint of the hypotenuse is equidistant from the 3 vertices.
    All I did with the slope calculation was to show that the triangle is indeed right-angled.
    Other than that, finding the slope had nothing to do with the calculations of the distances.

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    posted by Reiny

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