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.

Correct  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

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