The longest side of a right triangle has length 6. The triangle's area is also 6. If the perimeter of the triangle is [tex]a+b\sqrt{c}[/tex], where c is not divisible by the square of any prime, find a + b + c.
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Answer:
Nᴏᴡ, Hʏᴘᴏᴛᴇɴᴜsᴇ ɪs 6.
c² = a² + b²
⇒ 36 = a² + b² ___(1)
Now, there is a very interesting formula
a² + b² = (a + b)² - 2ab
Therefore, we can rearrange (1)
⇒ 36 = (a + b)² - 2ab___(2)
Area is 6.
___(3)
Put (3) in (2)
⇒ 36 = (a + b)² - 2(12)
⇒ 36 + 24 = (a + b)²
⇒ √60 = a + b
⇒ 2√15 = a + b
Now, a + b + c:
⇒ (2√15 + 6) cm.
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