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The diagram shows the triangle RST inscribed in the circle k - Junior Cycle Mathematics - Question 8 - 2015

Question 8

The diagram shows the triangle RST inscribed in the circle k. The line segment [RS] is a diameter of the circle. Gavin says: "The size of the angle W must be 90°." ... show full transcript

Worked Solution & Example Answer:The diagram shows the triangle RST inscribed in the circle k - Junior Cycle Mathematics - Question 8 - 2015

Step 1

State one result from your course (a theorem or a corollary) that shows that Gavin is correct.

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Answer

The angle at the centre of a circle is twice the angle at the circumference standing on the same arc [Theorem 19]. Alternatively, each angle in a semi-circle is a right angle [Corollary 3].

Step 2

Using this information, and trigonometry, find the size of LX.

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Answer

Given that |ST| = 10 and |RS| = 30, we can apply the sine rule.

We know that:

$ext{sin} X = \frac{|ST|}{|RS|} = \frac{10}{30}$

Therefore:

$X = \text{sin}^{-1}\left(\frac{10}{30}\right)$

Calculating this gives:

$X \approx 19.5°$

So, when rounded to one decimal place, $LX \approx 19.5°$ .

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