The lengths of two sides of a right triangle are 5 inches and 8 inches. what is the difference between the two possible lengths of the third side of the triangle? round your answer to the nearest tenth. inches

The difference between the two possible lengths of the third side 3.2 to the nearest tenth inches

Step-by-step explanation:

* Lets explain how to find a side in aright triangle from the other

 two sides

- In triangle ABC

# AB , BC , AC formed a right angle triangle  

- Their is the relation between the lengths of the 3 sides

# If AB is the longest side

∴ (AB)² = (BC)² + (AC)²

# If BC is the longest side

∴ (BC)² = (AB)² + (AC)²

# If AC is the longest side

∴ (AC)² = (AB)² + (BC)²

* Now lets solve the problem

- The lengths of two sides of a right triangle are 5 inches and 8 inches

- Let the length of the third side is x inches

- We don't know which on is the longest side

# First case

∵ 8 > 5, then let 8 is the longest side

∵ 8 is the longest side  

∴ (8)² = (5)² + (x)²

∴ 64 = 25 + x² ⇒ subtract 25 from both sides

∴ 39 = x² ⇒ take √ for both sides

∴ x = √39

∴ The length of the third side is √39 inches

# second case

- Let x is the longest side

∵ x is the longest side  

∴ (x)² = (5)² + (8)²

∴ x² = 25 + 64 ⇒ subtract 25 from both sides

∴ x² = 89 ⇒ take √ for both sides

∴ x = √89

∴ The length of the third side is √89 inches

- Lets find the difference between the two possible lengths of the

 third side

∵ The lengths of the third sides are √39 and √89

∴ The difference = √89 - √39 = 3.18898 ≅ 3.2

* The difference between the two possible lengths of the third side

 3.2 to the nearest tenth inches

you already gave the answer

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it is only things thain can be deafeated

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