Answer:
x = 4
y = 2√5
z = 3√5
Explanation:
There are 3 right triangles here, so we can create 3 expressions using Pythagoras' Theorem:
a² + b² = c²
(where a and b are the legs, and c is the hypotenuse)
smallest triangle: x² + y² = (x + 2)²
medium triangle: (x + 1)² + y² = z²
largest triangle: (x + 2)² + z² = (x + x + 1)²
rewrite the smallest triangle expression to make y² the subject:
x² + y² = (x + 2)²
⇒ x² + y² = x² + 4x + 4
⇒ y² = 4x + 4
rewrite the medium triangle expression to make z² the subject, substituting y² = 4x + 4:
(x + 1)² + y² = z²
⇒ (x + 1)² + 4x + 4 = z²
⇒ x² + 2x + 1 + 4x + 4 = z²
⇒ z² = x² + 6x + 5
rewrite the largest triangle, substituting z² = x² + 6x + 5 and solve for x:
(x + 2)² + z² = (x + x + 1)²
⇒ (x + 2)² + x² + 6x + 5 = (2x + 1)²
⇒ x² + 4x + 4 + x² + 6x + 5 = 4x² + 4x + 1
⇒ 2x² - 6x - 8 = 0
⇒ x² - 3x - 4 = 0
⇒ (x + 1)(x - 4) = 0
⇒ x = -1, x = 4
as x > 0, x = 4 only
Substitute found value for x into y² = 4x + 4 and solve for y:
y² = (4 x 4) + 4
⇒ y² = 20
⇒ y = √20 = 2√5
Substitute found value for x into z² = x² + 6x + 5 and solve for z:
z² = 4² + (6 x 4) + 5
⇒ z² = 45
⇒ z = √45 = 3√5