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10 September, 10:54

What is the length of the hypotenuse of ΔABC? (1) The lengths of the three sides of ΔABC are consecutive even integers. (2) The hypotenuse of ΔABC is 4 units longer than the shorter leg.

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  1. 10 September, 13:03
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    Correct answer: hypotenuse c = 10 units

    Step-by-step explanation:

    We will first define the variables:

    The definition of an even number is 2·n where n = 1,2,3,4, ...

    Let the shorter leg be b = 2·n, the longer leg a = 2· n + 2 and the hypotenuse c = 2· n + 4

    Write the Pythagorean Theorem:

    c² = a² + b²

    when we make the replacement we get

    (2· n + 4) ² = (2· n + 2) ² + (2· n) ² = > 4n² + 16n + 16 = 4n² + 8n + 4 + 4n² = >

    4n² - 8n - 12 = 0 when we divide equation by 4 we get

    n² - 2n - 3 = 0 = > n - 3n + n - 3 = > n (n - 3) + (n - 3) = (n - 3) · (n + 1) = 0 = >

    n = 3 or n = - 1

    We accept n = 3 and reject n = - 1 because n can only be a positive number

    when we return the replacement we get three sides of ΔABC

    c = 2· n + 4 = 2 · 3 + 4 = 10

    a = 2· n + 2 = 2 · 3 + 2 = 8

    b = 2·n = 2 · 3 = 6

    Check : c - b = 10 - 6 = 4 units

    God is with you!
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