Question

5 the diagram shows two similar quadrilaterals, (abcd) and (efgh)

(ab = 5mathrm{cm}) (bc = 4mathrm{cm}) (cd=ymathrm{cm})
(ef=xmathrm{cm}) (fg = 10mathrm{cm}) (gh = 24mathrm{cm})
(a) work out the value of (x)

(b) work out the value of (y)

Explanation:

Step1: Find the scale factor

Since the quadrilaterals are similar, the ratio of corresponding sides is equal. For part (a), we use the ratio of \(BC\) and \(FG\). The scale factor \(k=\frac{FG}{BC}\). Given \(BC = 4\mathrm{cm}\) and \(FG=10\mathrm{cm}\), \(k=\frac{10}{4}=\frac{5}{2}\).

Step2: Calculate \(x\)

\(AB\) and \(EF\) are corresponding sides. We know \(AB = 5\mathrm{cm}\). Using the scale - factor relationship \(\frac{EF}{AB}=\frac{FG}{BC}\), so \(x=\frac{FG\times AB}{BC}\). Substitute \(AB = 5\mathrm{cm}\), \(BC = 4\mathrm{cm}\), \(FG = 10\mathrm{cm}\) into the formula: \(x=\frac{10\times5}{4}=\frac{50}{4}=12.5\mathrm{cm}\).

Step3: Calculate \(y\)

For part (b), \(CD\) and \(GH\) are corresponding sides. Using the ratio \(\frac{GH}{CD}=\frac{FG}{BC}\). Let \(CD=y\), \(GH = 24\mathrm{cm}\), \(BC = 4\mathrm{cm}\), \(FG = 10\mathrm{cm}\). Then \(\frac{24}{y}=\frac{10}{4}\). Cross - multiply gives \(10y=24\times4\). So \(10y = 96\), and \(y=\frac{96}{10}=9.6\mathrm{cm}\).

Answer:

(a) \(x = 12.5\)
(b) \(y=9.6\)