The three sides of a right-angled triangle are x, x+1 and 5. find x and the area, if the
longest side is 5. the hypotenuse = 5 x + (x+1)2 = 52 (pythagoras' theorem)

see explanation

Step-by-step explanation:

Using Pythagoras' identity in the right triangle.

The square on the hypotenuse is equal to the sum of the squares on the other 2 sides, that is

x² + (x + 1)² = 5² ← expand factor and simplify left side

x² + x² + 2x + 1 = 25 ( subtract 25 from both sides )

2x² + 2x - 24 = 0 ← divide through by 2

x² + x - 12 = 0 ← in standard form

(x + 4)(x - 3) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 4 = 0 ⇒ x = - 4

x - 3 = 0 ⇒ x = 3

However, x > 0 ⇒ x = 3

The legs of the right triangle are 3 and x+ 1 = 3 + 1 = 4

The area (A) of the triangle is calculated as

A = 0.5bh ( b is the base and h the height )

Here b = 3 and h = 4, thus

A = 0.5 × 3 × 4 = 0.5 × 12 = 6 units²

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