The sides (in cm) of a right triangle are (x − 13), (x − 26) and x. Its area (in cm²) is:

Given:

Sides of right triangle: (x − 13), (x − 26), and x cm

Formula used:

Pythagoras Theorem: Hypotenuse² = Base² + Height²

Area = (1/2) × Base × Height

Calculation:

Hypotenuse = x, Base = (x - 26), Height = (x - 13)

Assume hypotenuse = x

⇒ x² = (x − 13)² + (x − 26)²

⇒ x² = (x² − 26x + 169) + (x² − 52x + 676)

⇒ x² = 2x² − 78x + 845

⇒ Bring all terms to one side:

⇒ x² − 2x² + 78x − 845 = 0

⇒ −x² + 78x − 845 = 0

⇒ x² − 78x + 845 = 0

Solve using quadratic formula:

x = [78 ± √(78² − 4 × 1 × 845)] ÷ 2

x = [78 ± √(6084 − 3380)] ÷ 2

x = [78 ± √2704] ÷ 2

x = [78 ± 52] ÷ 2

⇒ x = (78 + 52)/2 = 130 ÷ 2 = 65 (valid)

⇒ x = (78 − 52)/2 = 26 ÷ 2 = 13 (invalid as sides become 0)

If x = 13, then one side is (x - 13) = (13 - 13) = 0, which is not possible for a triangle. So, x ≠ 13.

So, x = 65

Now, find the lengths of the sides:

Side 1 = x - 13 = 65 - 13 = 52 cm

Side 2 = x - 26 = 65 - 26 = 39 cm

Hypotenuse = x = 65 cm

Area = (1/2) × 39 × 52 = 1014 cm²

∴ Area = 1014 cm²