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Two triangles are drawn on a square grid as shown - Leaving Cert Mathematics - Question 4 - 2011
Question 4
Two triangles are drawn on a square grid as shown. The points P, Q, R, X, and Z are on vertices of the grid, and the point Y lies on PR. The triangle PQR is an enlar... show full transcript
Worked Solution & Example Answer:Two triangles are drawn on a square grid as shown - Leaving Cert Mathematics - Question 4 - 2011
Step 1
Calculate the scale factor of the enlargement, showing your work.
Answer
To find the scale factor of the enlargement, we need to compare the lengths of corresponding sides of the triangles PQR and XYZ.
Let's calculate the lengths:
- The length |PR| is given as 6 units.
- The length |XZ| is given as 4 units.
The scale factor (k) is calculated as:
Thus, the scale factor of the enlargement is ( \frac{3}{2} ).
Step 2
By construction or otherwise, locate the centre of enlargement on the diagram above.
Answer
The centre of enlargement can be found by drawing lines from each point of triangle XYZ through its corresponding points in triangle PQR. The intersection of these lines will give us the centre of enlargement.
As shown in the diagram above, the centre of enlargement is located at point O.
Step 3
Calculate |YR| in grid units.
Answer
To calculate the length |YR|, we can apply the property of the enlargement:
The corresponding lengths between the two triangles are related by the scale factor, which we found to be ( \frac{3}{2} ). The formula for the corresponding side length |YR| in terms of side |XY| is:
From the diagram, we can determine:
- |QR| = 8 units (using the triangle side lengths).
Thus,
Therefore, the value of |YR| in grid units is ( \frac{16}{3} \text{ units} ).