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. 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 Constructions: 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

In square ABCD:

Each angle is $90^ ext{o}$ . Therefore,
Angle $W = 60^ ext{o}$ (from the equilateral triangle DCE).
Angle $X = 90^ ext{o}$ (as it's a right angle in the square).
Angle $Y = 45^ ext{o}$ (since angles in triangle BCA are complementary).

Step 2

Work out the size of the angle Z.

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Answer

Angle Z is obtained by adding the angles W and Y: $Z = W + Y = 60^ ext{o} + 45^ ext{o} = 105^ ext{o}$ .

Step 3

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

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Answer

In the square: Using the theorem of Pythagoras: $x^2 = 5^2 + 5^2$ $x^2 = 25 + 25 = 50$ $x = \sqrt{50} = 7.071$ Thus, the length of the diagonal is approximately $7.07$ cm (to two decimal places).

Step 4

Construct this diagram in the space below.

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Answer

Draw a square ABCD with each side measuring 5 cm, and an equilateral triangle DCE on top of side DC with each side measuring 5 cm.

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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