In triangle \(XYZ\), angle \(XYZ=90^\circ\), angle \(ZXY=37^\circ\) and \(XY=12.5\) cm. Calculate the length of \(YZ\), correct to \(3\) significant figures...
Question 1 Report
In triangle \(XYZ\), angle \(XYZ=90^\circ\), angle \(ZXY=37^\circ\) and \(XY=12.5\) cm.
Calculate the length of \(YZ\), correct to \(3\) significant figures.
Answer Details
The right angle is at \( Y \), and the known angle is at \( X \). Relative to the \( 37^\circ \) angle, \( YZ \) is the opposite side and \( XY = 12.5 \) cm is the adjacent side, so tangent is the correct ratio.
\[ \tan 37^\circ = \frac{YZ}{12.5} \]Rearranging to make \( YZ \) the subject:
\[ YZ = 12.5 \times \tan 37^\circ \] [M1] \[ = 12.5 \times 0.753554\ldots = 9.4194\ldots \]To \( 3 \) significant figures, \( YZ = 9.42 \) cm. [A1]
Because \( 37^\circ \) is less than \( 45^\circ \), \( \tan 37^\circ \lt 1 \), so \( YZ \) must be shorter than \( XY \), and \( 9.42 \lt 12.5 \) as expected. Dividing by \( \tan 37^\circ \) instead of multiplying would give \( 16.6 \) cm, which fails that check.
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