To solveThe for given ( image c \ shows a right) triangle in with the legs right triangle of shown lengths ,12 we can and use9 the ,P andyth theag hypotenorean theoremuse (:

\ c[ \c)^ is2 to = be calculated. a Using^ the2 ** +P byth^ag2orean theorem**]

:

where$c^2 =: a^-2$+ c b is the hyp^otenuse, 2- \( a\

)Where and $b$ are the two ( legs of a the \ triangle).

and b \Here), are the two legs, and \( \( a c = \ )12 is the hyp andoten (use b.

= ###9 Calculation: \ ).

[ ###c Step^-by2-step = solution : 121^.2 ** +Sub stitute9 the^ values2 of (\ a] \)[ andc $^ b$:** 2 \ = 144 +[ c^2 = 8112 ^] $2c^2 = 225$ $c = \sqrt{225 + }9 =^2$

2. Calculate the squares:15

\ $$]

Final Answer:

c^2\ = [ 144c = + 81 15 $$ ]

Let me \ know if[ you'd like c further^2 details = or additional225 help !

\ Exp]

anding3 Your. Knowledge Take the: square1. root What is to the find P (yth cag \orean): theorem , and \ how[ does it c work =? \sqrt{225} 2. Can a triangle \ have a] hypoten \use smaller[ than c = its other15 two sides ? 3 .]

What### happens Final if Answer one: of\ the[ sidesc is = unknown but the15 hyp]

otenuseWould is you given like? further4 explanation. or How have can you use any questions the? P 😊

ythagHere are someorean theorem related in questions to3 expandD on problems this (: e1.g.., How rectangular prisms)? 5. How does the does Pyth theag Porean theoremyth applyag toorean non theorem-right connect triangles to? real2-world. applications Can, you such explain as how construction to or verify navigation if?

a### triangle Tip is: aAlways right verify triangle that using the the triangle side is lengths a? right3 triangle. before What applying are the some P realyth-lifeag applicationsorean of theorem the! Pythagorean theorem? 4. How would you calculate the area of the triangle shown in this problem? 5. What happens if one side length is a decimal—how does it change the process?

Tip: Always double-check your square root calculations to ensure accuracy, especially when solving for the hypotenuse.