Quadratic applied to similar/geometric figures with surds

A question is this type if and only if it involves similar shapes or geometric figures with surd-valued dimensions, requiring multiplication or simplification of surds to find an unknown length.

2 questions · Moderate -0.1

1.02b Surds: manipulation and rationalising denominators
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Edexcel C12 2017 October Q4
6 marks Moderate -0.5
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-08_287_689_255_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C = 24 \sqrt { 3 }\)
  1. show that \(x = 4 \sqrt { 2 }\)
  2. Hence find the exact length of \(B C\), giving your answer as a simplified surd.
OCR C1 Q3
5 marks Standard +0.3
3. \includegraphics[max width=\textwidth, alt={}, center]{4fec0924-d727-4d4f-81e6-918e1ccfedbd-1_330_1230_829_386} The diagram shows the rectangles \(A B C D\) and \(E F G H\) which are similar.
Given that \(A B = ( 3 - \sqrt { 5 } ) \mathrm { cm } , A D = \sqrt { 5 } \mathrm {~cm}\) and \(E F = ( 1 + \sqrt { 5 } ) \mathrm { cm }\), find the length \(E H\) in cm, giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are integers.