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Answer on Geometry Question for Elizabeth Mobley

Question #6279
If the legs of a triangle are 7 cm and the base is 5 cm, what is the area?
Expert's answer
Consider triangle ABC:
data:image/png;base64,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
We know that AB=BC=7 cm, AC=5cm.
Let's draw height BD. As long as AC=BC , the height BD is also a median: AD=DC=AC/2=2.5
Consider triangle BDC, its right angled. Let's find BD using Pethagorean theorem:
BD^2+DC^2=BC^2
Hence BD=sqrt(BC^2-DC^2)=sqrt(7^2-2.5^2)= approx. 6.54
Now we can find area:
S=1/2*AC*BD=1/2*5*6.54=16,35 cm2

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