recall that an isosceles triangle has two equal sides. thus, to find this length, recall that the perimeter of a triangle is given by
P = a + b + c
but since two sides are equal, we can rewrite this as
P = 2a + c
32 = 2a + 12
20 = 2a
a = 10
now to find the area, if you draw the isosceles triangle and draw the height, you actually form two right triangles. we need to find this height, and to do this, we divide the base by 2, and use the pythagorean theorem:
c^2 = a^2 + b^2
note that c is the hypotenuse which is equal to 10, and 12/2 = 6 is a. (try to imagine or draw the figure)
10^2 = 6^2 + b^2
100 = 36 = b^2
64 = b^2
b = 8 cm (height)
finally, recall that area of a triangle is given by
A = (1/2)bh
where b is the base, and h is the height. substituting,
A = (1/2)(12)(8)
A = 6*8
A = 48 cm^2
hope this helps~ :)
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