Answer: Approximately 1336.50301206033 square cm
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Explanation:
If we know the three side lengths of a triangle, then we can use Heron's formula to get the area
Heron's formula is
[tex]A = \sqrt{S(S-a)(S-b)(S-c)[/tex]
with S being the semiperimeter, or half the perimeter, so [tex]S = \frac{a+b+c}{2}[/tex]. We add up all three sides and divide by 2.
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The smaller triangle has side lengths of a = 19, b = 40, c = 41. So we get a semiperimeter of S = (a+b+c)/2 = (19+40+41)/2 = 50
Plug this along with the a,b,c values to get an area of...
[tex]A = \sqrt{S(S-a)(S-b)(S-c)}\\\\A = \sqrt{50(50-19)(50-40)(50-41)}\\\\A = \sqrt{139500}\\\\A \approx 373.496987939661\\\\[/tex]
which is approximate
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Do the same for the larger triangle as well
S = (a+b+c)/2 = (19+180+181)/2 = 190
[tex]A = \sqrt{S(S-a)(S-b)(S-c)}\\\\A = \sqrt{190(190-19)(190-180)(190-181)}\\\\A = \sqrt{2924100}\\\\A = 1710\\\\[/tex]
Interestingly enough, we get an exact area value this time as opposed to some approximation.
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The difference between the two results is the area of the shaded region
1710 - 373.496987939661 = 1336.50301206033
This is approximate since the first result was approximate.
