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 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!