How to Find the Area of a Rectangle Using the Diagonal
Let the diagonal = c^2 and let the height and width of the rectangle = a^2 and b^2, respectively. , Let the diagonal = 225, and let its square = 60,025; you know that a Pythagorean Triple is 3-4-5. with the hypotenuse equal to 5, so now you want to...
Step-by-Step Guide
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Step 1: Let the diagonal = c^2 and let the height and width of the rectangle = a^2 and b^2
Since 3 and 4 are close to each other, try a number between 3^2 and 4^2, like
12.5 and various x's to the 4th power.
Try
6. 6^4 *
12.5 = 16,228.512 which is not bad, considering we averaged, but it's low., You did it! 3^2 = 9 and 4^2 = 16 and x^4 = 7^4., 147^2 + 196^2 = 60,025 = c^2.
Its square root is 225, the diagonal., However, there are various other triples, since all that is required of the triangle is that it have a 90 degree angle in it.
Still, with diligence, one may discover the given relation for a given rectangle by close observation of how long and thin it is.
Direct relative measurement of the sides of the rectangle also helps in finding its area, but error can creep in with direct measurement so formulation is needed to provide a more exact answer, typically. -
Step 2: respectively.
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Step 3: Let the diagonal = 225
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Step 4: and let its square = 60
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Step 5: 025; you know that a Pythagorean Triple is 3-4-5. with the hypotenuse equal to 5
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Step 6: so now you want to know how to get the other sides
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Step 7: given that information.
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Step 8: Reason the answer is going to be =(3*x^2)^2+(4*x^2)^2
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Step 9: or 3^2x^4 +4^2x^4.
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Step 10: Try 7 because 60
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Step 11: 025/5^2=7^4=2401: 7^4 *9 + 7^4 *16 = 60
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Step 12: 025 -- the square of the diagonal!
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Step 13: Learn that the shorter side is sqrt(3* 7^2)^2 = 147
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Step 14: and the longer side is sqrt(4 * 7^2)^2 = 196.
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Step 15: Deduce that the area of the rectangle then is 196 * 147 = 28
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Step 16: 812 You are warned
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Step 17: however
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Step 18: that this method works because many Pythagorean Triples are related to the 3-4-5 triple
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Step 19: but not all.
Detailed Guide
Since 3 and 4 are close to each other, try a number between 3^2 and 4^2, like
12.5 and various x's to the 4th power.
Try
6. 6^4 *
12.5 = 16,228.512 which is not bad, considering we averaged, but it's low., You did it! 3^2 = 9 and 4^2 = 16 and x^4 = 7^4., 147^2 + 196^2 = 60,025 = c^2.
Its square root is 225, the diagonal., However, there are various other triples, since all that is required of the triangle is that it have a 90 degree angle in it.
Still, with diligence, one may discover the given relation for a given rectangle by close observation of how long and thin it is.
Direct relative measurement of the sides of the rectangle also helps in finding its area, but error can creep in with direct measurement so formulation is needed to provide a more exact answer, typically.
About the Author
Zachary Harris
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