A festival crew is fencing off a triangular area beside the main stage for equipment storage overnight. Two sides of the triangle measure \(\sqrt{45}\) metr...
Question 1 Report
A festival crew is fencing off a triangular area beside the main stage for equipment storage overnight. Two sides of the triangle measure \(\sqrt{45}\) metres and \(\sqrt{20}\) metres.
Show that the total length of these two sides can be written as \(k\sqrt{5}\) metres, where \(k\) is an integer, and find the value of \(k\). (3)
Answer Details
To write each surd in the form \(k\sqrt{5}\), find the largest factor of each number under the root that is a perfect square multiple of \(5\). Since \(45=9\times5\), \(\sqrt{45}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}\). [1 mark]
Since \(20=4\times5\), \(\sqrt{20}=\sqrt{4}\times\sqrt{5}=2\sqrt{5}\). [1 mark]
Now that both lengths share the same \(\sqrt{5}\) part, they can be added like like terms: \(3\sqrt{5}+2\sqrt{5}=5\sqrt{5}\) metres, so \(k=5\). [1 mark]
Surds with different numbers under the root, \(\sqrt{45}\) and \(\sqrt{20}\), can only be combined by addition once each is broken down to reveal a shared surd part, here \(\sqrt{5}\).
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