Solve for the area of this triangle, explanation would be greatly appreciated!

9514 1404 393

  13√731 ≈ 351.48 cm²

Step-by-step explanation:

When given three sides of a triangle, there are a number of ways that the area may be calculated. Perhaps the most straightforward is the use of Heron's formula.

Where s is the semi-perimeter, the area is given by ...

  A = √(s(s -a)(s -b)(s -c)) . . . . . . . . s = (a+b+c)/2

Here, the value of s is ...

  s = (30 +30 +26)/2 = 43

  A = √(43(13)(13)(17)) = 13√(43·17)

  A = 13√731 ≈ 351.48 . . . . . square centimeters

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Additional comments

This triangle is isosceles, so the altitude can be found from the Pythagorean theorem:

  h² +13² = 30²

  h = √(900 -169) = √731

Then the area is ...

  A = (1/2)bh = (1/2)(26 cm)(√731 cm) = 13√731 cm²

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You can also find one of the angles using the Law of Cosines, then find the area from ...

  Area = (1/2)a·b·sin(C) . . . . . . for any two sides a, b, and enclosed angle C

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Ultimately, all of these methods are equivalent to Heron's formula.