The triangle PQR is the image of the triangle DEF under an enlargement. (The diagram is not drawn to scale.) The scale factor of the enlargement, k, is 2:5. |EF| = 3 cm. (i) Use the scale factor to find |QR|. (ii) The area of the triangle PQR is 18.75 cm². Find the area of triangle DEF.

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The triangle PQR is the image of the triangle DEF under an enlargement - Leaving Cert Mathematics - Question b - 2021

Question b

The triangle PQR is the image of the triangle DEF under an enlargement. (The diagram is not drawn to scale.) The scale factor of the enlargement, k, is 2:5. |EF| = 3... show full transcript

Worked Solution & Example Answer:The triangle PQR is the image of the triangle DEF under an enlargement - Leaving Cert Mathematics - Question b - 2021

Step 1

Use the scale factor to find |QR|.

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Answer

To find |QR|, we can use the scale factor given:

$|QR| = |EF| \times k$

Since |EF| = 3 cm and the scale factor k = \frac{2}{5}, we can substitute these values into the formula:

$|QR| = 3 \times \frac{2}{5}$

Calculating this gives us:

$|QR| = \frac{6}{5} cm = 1.2 cm$

Step 2

The area of triangle DEF.

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Answer

Given that the area of triangle PQR is 18.75 cm², we can find the area of triangle DEF using the ratio of the areas which is the square of the scale factor:

$\text{Area ratio} = k^2 = \left(\frac{2}{5}\right)^2 = \frac{4}{25}$

Now, we can calculate the area of triangle DEF:

$\text{Area}_{DEF} = \text{Area}_{PQR} \times \text{Area ratio}$

Substituting the known values gives us:

$\text{Area}_{DEF} = 18.75 \times \frac{4}{25} = 3 cm²$

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