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A and B are points on a circle with centre C - Junior Cycle Mathematics - Question 4 - 2012

Question 4

A and B are points on a circle with centre C. ∠|BCA| = 44°. Find ∠|BAC|. XYZ is a right angled triangle with |∠XYZ| = 90°. W is a point on |XZ|, such that |YW| is... show full transcript

Worked Solution & Example Answer:A and B are points on a circle with centre C - Junior Cycle Mathematics - Question 4 - 2012

Step 1

Find ∠|BAC|

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Answer

To find ∠|BAC|, use the property of angles in a circle. The angle at the center (∠|BCA|) is twice the angle at the circumference (∠|BAC|).

Using the formula: $ext{∠|BAC|} = rac{1}{2} (180° - ∠|BCA|)$

Substituting $ext{∠|BAC|} = rac{1}{2} (180° - 44°) = rac{1}{2} (136°) = 68°$

So, ∠|BAC| = 68°.

Step 2

Find a in the triangle XYZ

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Answer

Given that W is on |XZ| and |YW| is perpendicular to |XZ|, we can consider the properties of similar triangles.

From the right triangle YWZ, we can use the Pythagorean theorem: $|YW|^2 + |WZ|^2 = |YZ|^2$

Substituting the known values: $(3a)^2 + a^2 = 16^2$ $9a^2 + a^2 = 256$ $10a^2 = 256$ $a^2 = 25.6$ $a = rac{ ext{\sqrt256}}{ ext{\sqrt10}} = 8.0$

Thus, the value of a is 8 cm.

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