In the diagram below, ABCD is a square and DCE is an equilateral triangle. Some of the angles are marked. (i) Find the size of the angles W, X, and Y. Z is the obtuse angle ACE. (ii) Work out the size of the angle Z. (b) The square and the equilateral triangle in the diagram have sides of length 5 cm, as shown on the right. (i) Use the theorem of Pythagoras to find the value of x, the length of the diagonal of the square. Give your answer correct to two decimal places. (ii) Construct this diagram in the space below.

Junior Cycle Mathematics Trigonometry: In the diagram below, ABCD is a

In the diagram below, ABCD is a square and DCE is an equilateral triangle - Junior Cycle Mathematics - Question 9 - 2018

Question 9

In the diagram below, ABCD is a square and DCE is an equilateral triangle. Some of the angles are marked. (i) Find the size of the angles W, X, and Y. Z is the obt... show full transcript

Worked Solution & Example Answer:In the diagram below, ABCD is a square and DCE is an equilateral triangle - Junior Cycle Mathematics - Question 9 - 2018

Step 1

Find the size of the angles W, X, and Y.

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Answer

To find the angles W, X, and Y, we utilize the properties of the square and triangle:

In square ABCD, each angle is 90°.
Therefore, we have:
- $|W| = 90° - 30° = 60°$ (since angle EDC in triangle DCE is 60°)

Thus, the angles are:

$|W| = 60°$
$|X| = 90°$ (right angle at B)
$|Y| = 45°$ (as both angles at A are equal due to triangle symmetry).

Step 2

Work out the size of the angle Z.

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Answer

To find the angle Z (the obtuse angle ACE), we apply the following:

The sum of angles in triangle DCE is 180°.
Given that CE is equilateral, we already have:
- $|DCE| = 60°$
- $|E| + |D| = 120°$

So:

$|Z| = 180° - (|W| + |X|) = 180° - (60° + 45°) = 105°$ .

Step 3

Use the theorem of Pythagoras to find the value of x.

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Answer

In triangle formed by the diagonal (x), we apply Pythagoras' theorem:

Square the lengths: $x^2 = 5^2 + 5^2$
Thus, we have: $x^2 = 25 + 25 = 50$
Taking the square root gives: $x = \sqrt{50} = 7.071...\ (correct \ to \ 2 \ D.P = 7.07)$ .

Step 4

Construct this diagram in the space below.

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Answer

When constructing the diagram, ensure that:

The square ABCD is accurately represented with each side measuring 5 cm.
The equilateral triangle DCE has all sides also measuring 5 cm.
The diagonal AC is drawn to depict its length x cm. Make sure to label angles and sides clearly.

In the diagram below, ABCD is a square and DCE is an equilateral triangle - Junior Cycle Mathematics - Question 9 - 2018

Worked Solution & Example Answer:In the diagram below, ABCD is a square and DCE is an equilateral triangle - Junior Cycle Mathematics - Question 9 - 2018

Find the size of the angles W, X, and Y.

Work out the size of the angle Z.

Use the theorem of Pythagoras to find the value of x.

Construct this diagram in the space below.

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