How to Find \(\sqrt{9.3}\) Geometrically
Learn how to find the value of \(\sqrt{9.3}\) with a simple geometric construction. Follow these easy steps to see math in action!
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Draw the Base Line
Draw a line segment AB of 9.3 units. From point B, extend it 1 unit to point C.
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Make a Semicircle
Find the middle of AC and call it O. With O as the center, draw a semicircle from A to C.
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Draw a Vertical Line
From point B, draw a straight line up until it hits the semicircle. Call this point D.
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Find the Root!
The length of the line BD is your answer. So, BD = \(\sqrt{9.3}\).
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Mark it on the Line
With a compass point on B and pencil on D, swing down to the main line to make point E. The length BE is \(\sqrt{9.3}\).
💡 The Math Behind the Method
This trick works for any number, not just 9.3! If you call the starting length \(x\), the construction creates a right-angled triangle (\(\triangle OBD\)).
Using the Pythagorean Theorem, the sides are related like this:
\[ BD^2 = (\text{radius } OD)^2 - (\text{side } OB)^2 \]The math cleverly simplifies so that in the end, you always get:
\[ BD^2 = x \] \[ BD = \sqrt{x} \]That's why the length of BD is the exact square root you were looking for!
