We are given a spiral of right triangles, each built on the hypotenuse of the previous one. All the outer legs (the ones not shared) are length 1. We need to find x, which is the hypotenuse of the fifth triangle.
Let’s go step by step, as the hint suggests: find c₁, then c₂, c₃, c₄, and finally x.
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Step 1: Find c₁
The first triangle has two legs: both are 1.
Using the Pythagorean theorem:
c₁² = 1² + 1² = 1 + 1 = 2
→ c₁ = √2
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Step 2: Find c₂
The second triangle has one leg = 1 (outer), and the other leg = c₁ = √2.
So:
c₂² = 1² + (√2)² = 1 + 2 = 3
→ c₂ = √3
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Step 3: Find c₃
Third triangle: legs are 1 and c₂ = √3
c₃² = 1² + (√3)² = 1 + 3 = 4
→ c₃ = √4 = 2
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Step 4: Find c₄
Fourth triangle: legs are 1 and c₃ = 2
c₄² = 1² + 2² = 1 + 4 = 5
→ c₄ = √5
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Step 5: Find x
Fifth triangle: legs are 1 and c₄ = √5
x² = 1² + (√5)² = 1 + 5 = 6
→ x = √6
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Final Answer:
√6
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem worksheet word problems.
