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

Given the length of |AE|, we can apply similar triangles to find |AD|.

Using the ratio from the triangles:

$\frac{|AD|}{|AE|} = \frac{12}{5}$

Now substituting the known value of |AE|:

$|AD| = |AE| \times \frac{12}{5}$

Therefore:

$|AD| = 12 \times \frac{12}{5} = \frac{144}{5} = 31.2 \, cm$

To show that |AB| equals 36 cm, we will use the properties of the similar triangles ΔAFE and ΔAGB.

Since |EF| corresponds to |GB|, we can set up our proportion based on the previously established similarity:

1. From ΔAFE and ΔAGB, we know:

   • (\frac{|EF|}{|AG|} = \frac{|AF|}{|AB|})
   • Given |EF| = 5 cm and |AG| = 39 cm, we find:

   \frac{5}{39} = \frac{|AF|}{|AB|}\

2. Let’s express |AB| using |AF|, which is already known:

   • |AF| = 12 cm, thus:

   \frac{5}{39} = \frac{12}{|AB|}\

Cross-multiplying gives:

$5 \cdot |AB| = 39 \cdot 12$

3. Solving for |AB|:

   $|AB| = \frac{39 \cdot 12}{5} = 36 \, cm$

To find the area of quadrilateral GCDE, we can use the fact that it consists of a rectangle and a triangle.

1. Since ABCD is a rectangle, the area of rectangle ABCD is: $\text{Area} = |AB| \times |AD| = 36 \, cm \times 31.2 \, cm$

2. The area of triangle AFE (part of GCDE): $\text{Area}_{AFE} = \frac{1}{2} \times |AF| \times |EF| = \frac{1}{2} \times 12 \, cm \times 5 \, cm$ = 30 cm²

3. Thus: $\text{Area of GCDE} = \text{Area}_{ABCD} - \text{Area}_{AFE}$ = (36 \times 31.2) - 30 = 1122.2 - 30 = 1092.2 , cm²