Question

1. \\(\triangle abc\\) is similar to \\(\triangle xyz\\). each side of \\(\triangle xyz\\) is larger than the corresponding side of \\(\triangle abc\\) by a factor of 5.1. if the perimeter of \\(\triangle abc\\) is 20 cm, determine the perimeter of \\(\triangle xyz\\). 4 2. \\(\triangle hij\\) is similar to \\(\triangle klm\\). each side of \\(\triangle klm\\) is larger than the corresponding side of \\(\triangle hij\\) by a factor of 3. if the area of \\(\triangle hij\\) is 12 \\(cm^{2}\\), determine the area of \\(\triangle klm\\). 4

Explanation:

Step1: Recall the property of perimeters of similar - triangles

For two similar triangles, if the scale factor of side - lengths is \(k\), the ratio of their perimeters is equal to the scale factor.
Let the perimeter of \(\triangle ABC\) be \(P_{ABC}\) and the perimeter of \(\triangle XYZ\) be \(P_{XYZ}\), and the scale factor \(k = 5.1\). We know that \(\frac{P_{XYZ}}{P_{ABC}}=k\). Given \(P_{ABC}=20\mathrm{cm}\).
\[P_{XYZ}=k\times P_{ABC}\]

Step2: Calculate the perimeter of \(\triangle XYZ\)

Substitute \(k = 5.1\) and \(P_{ABC}=20\mathrm{cm}\) into the formula.
\[P_{XYZ}=5.1\times20 = 102\mathrm{cm}\]

Step3: Recall the property of areas of similar - triangles

For two similar triangles, if the scale factor of side - lengths is \(k\), the ratio of their areas is \(k^{2}\). Let the area of \(\triangle HIJ\) be \(A_{HIJ}\) and the area of \(\triangle KLM\) be \(A_{KLM}\), and the scale factor \(k = 3\). We know that \(\frac{A_{KLM}}{A_{HIJ}}=k^{2}\). Given \(A_{HIJ}=12\mathrm{cm}^{2}\).
\[A_{KLM}=k^{2}\times A_{HIJ}\]

Step4: Calculate the area of \(\triangle KLM\)

Substitute \(k = 3\) and \(A_{HIJ}=12\mathrm{cm}^{2}\) into the formula.
\[A_{KLM}=3^{2}\times12=9\times12 = 108\mathrm{cm}^{2}\]

Answer:

1. The perimeter of \(\triangle XYZ\) is \(102\mathrm{cm}\).
2. The area of \(\triangle KLM\) is \(108\mathrm{cm}^{2}\).