SOLUTION: GIVEN A TRIANGLE ABC WHERE THE VALUE OF BC^2 =370 UNITS AC^2 =74 AND AB^2=116 CALCULATE THE AREA OF A TRIANGLE / HINT; 370=9^2+17^2,74 = 5^2+7^2 AND 116 =4^2+10^2

Question 461155: GIVEN A TRIANGLE ABC WHERE THE VALUE OF BC^2 =370 UNITS AC^2 =74 AND AB^2=116 CALCULATE THE AREA OF A TRIANGLE /
HINT; 370=9^2+17^2,74 = 5^2+7^2 AND 116 =4^2+10^2
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Please refrain from caps lock...it gives the impression you are shouting your question.

We could attempt to find one of the angles using the law of cosines, then find the area using two sides and the in-between angle. However, this gets ugly, plus there is a way to take advantage of the "hint."

We can plot the triangle on a Cartesian plane instead, such that it satisfies the requirements:

We see that, from the Pythagorean theorem, AC^2 = 7^2 + 5^2, etc.

To find the area of ABC, create a rectangle with sides parallel to the axes and with diagonal BC:

Next, we just subtract off areas from the big triangle. The area (denote by [ABC]) of triangle ABC is:

The coordinates are all lattice points, so the areas are easy to evaluate.

(square units)

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