The diagram shows a rectangle, ABDE, and two congruent triangles, AFE and BCD - Edexcel - GCSE Maths - Question 15 - 2019 - Paper 3

The area of rectangle ABDE can be expressed as:

$$\text{Area}_{ABDE} = AB \cdot AE$$

Given that AB : AE = 1 : 3, let us denote AB = x. Then, AE = 3x. Therefore, we can express the area of the rectangle as:

$$\text{Area}_{ABDE} = x \cdot 3x = 3x^2$$

Now, we set the area of rectangle ABDE equal to the sum of the areas of triangles AFE and BCD:

$$3x^2 = 144 + 144 ext{ (since areas of triangles AFE and BCD are equal)}$$

ores

$$3x^2 = 288$$

Dividing both sides by 3 gives:

$$x^2 = 96$$

Taking the square root, we find:

$$x = \sqrt{96} = 4\sqrt{6} ext{ cm}$$

Since AE = 3x, we have:

$$AE = 3 \cdot 4\sqrt{6} = 12\sqrt{6} ext{ cm}$$

Thus, the length of AE is approximately 29.39 cm when evaluated.