Part of the logo of the company is shown below. ABCD is a square, with sides of length 30 cm. The points E and F are the midpoints of [AB] and [AD], respectively. The lines EC and FB are perpendicular, and meet at the point O. Using similar triangles or trigonometry, find the length |EO|. Give your answer in the form a√D cm, where a, b ∈ ℕ.

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Part of the logo of the company is shown below - Leaving Cert Mathematics - Question e - 2022

Question e

Part of the logo of the company is shown below. ABCD is a square, with sides of length 30 cm. The points E and F are the midpoints of [AB] and [AD], respectively. Th... show full transcript

Worked Solution & Example Answer:Part of the logo of the company is shown below - Leaving Cert Mathematics - Question e - 2022

Step 1

Determine |EB|

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Answer

Since E is the midpoint of AB, we have:

$|EB| = \frac{30}{2} = 15 \text{ cm}$

Step 2

Determine |EC|

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Answer

Next, using the Pythagorean theorem in triangle AEO, we have:

$|EC| = \sqrt{|EA|^2 + |AC|^2} = \sqrt{15^2 + 30^2} = \sqrt{225 + 900} = \sqrt{1125} = 15\sqrt{5} \text{ cm}$

Step 3

Use Similar Triangles

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Answer

Triangles ABE and ABC are similar, so we can set up the following ratio:

$\frac{|EO|}{|EB|} = \frac{|EC|}{|AB|}$

Substituting the known values:

$\frac{|EO|}{15} = \frac{15\sqrt{5}}{30}$

Step 4

Solve for |EO|

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Answer

Cross-multiplying gives:

$|EO| = 15 \times \frac{15\sqrt{5}}{30} = \frac{225\sqrt{5}}{30} = 7.5\sqrt{5}$

Thus, writing in the required form, we have:

$|EO| = 7.5\sqrt{5} \text{ cm}$

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