ABCD is a rectangle - Leaving Cert Mathematics - Question 5 - 2017

To find |AD|, we can use the properties of rectangles. The length |AD| can be determined using the Pythagorean theorem:

Given |AE| = 12 cm and |FE| = 5 cm:

1. Calculate |AF|:

   $|AF| = |AE| + |EF| = 12 + 5 = 17 cm.$

2. Calculate |AD| using the length of |EG| which is 27 cm:

   $|AD| = |AF| + |EG| = 17 + 27 = 31 cm.$

Using the similarity of triangles ΔAFE and ΔAGB:

We know the ratio of the sides corresponding to similar triangles:

1. From the similarity, we define the ratios:

   • $\frac{|AE|}{|AG|} = \frac{|AF|}{|AB|}$
   • Where |AE| = 12 cm, |AG| = 27 cm, and |AF| = 17 cm.

2. Solving for |AB| gives us:

   $\frac{12}{27} = \frac{17}{|AB|}$

   Thus, cross-multiplying we find:
   $12 |AB| = 17 \cdot 27$

   Simplifying:
   $|AB| = \frac{17 \cdot 27}{12} = 36 cm.$

To find the area of quadrilateral GCDE, we can break it down into triangles. The area can be calculated as:

1. Recognizing that GCDE forms parts of a rectangle, we calculate:

   • Area of quadrilateral = Area of rectangle - Area of two triangles (AFE & AGB).

2. As rectangles have an area calculated by length multiplied by width, we calculate:

   $Area = |AB| \cdot |AD| = 36 \cdot 31 = 1116 cm^2.$