In the right-angled triangle shown in the diagram, one of the acute angles is four times as large as the other acute angle. (i) Find the measures of the two acute angles in the triangle. (ii) The triangle in part (i) is placed on a co-ordinate diagram. The base is parallel to the x-axis. Find the slope of the line l that contains the hypotenuse of the triangle. Give your answer correct to three decimal places.

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In the right-angled triangle shown in the diagram, one of the acute angles is four times as large as the other acute angle - Junior Cycle Mathematics - Question 17 - 2014

Question 17

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In the right-angled triangle shown in the diagram, one of the acute angles is four times as large as the other acute angle. (i) Find the measures of the two acute a... show full transcript

Worked Solution & Example Answer:In the right-angled triangle shown in the diagram, one of the acute angles is four times as large as the other acute angle - Junior Cycle Mathematics - Question 17 - 2014

Step 1

Find the measures of the two acute angles in the triangle.

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Answer

Let the smallest angle be denoted as $x$ . Then, the other acute angle is $4x$ . Since the sum of the angles in a triangle is $180^{ ext{°}}$ , we have:

$ext{Sum of angles} = x + 4x + 90^{ ext{°}} = 180^{ ext{°}}$

Simplifying this gives:

$5x + 90 = 180$

Subtracting $90$ from both sides:

$5x = 90$

Dividing by $5$ yields:

$x = 18^{ ext{°}}$

Thus, the other acute angle is:

$4x = 4 \times 18 = 72^{ ext{°}}$

Therefore, the two acute angles are $18^{ ext{°}}$ and $72^{ ext{°}}$ .

Step 2

Find the slope of the line l that contains the hypotenuse of the triangle.

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Answer

In order to find the slope of the line $l$ that represents the hypotenuse of the triangle, we need to use the angle we found in part (i). The angle of the line with respect to the base is $18^{ ext{°}}$ . The slope ( $m$ ) is given by:

$m = an( heta)$

where $heta$ is the angle with the horizontal. Thus:

$m = an(18^{ ext{°}}) \\ m \approx 0.325$

So, the slope of the line $l$ is approximately $0.325$ , correct to three decimal places.

In the right-angled triangle shown in the diagram, one of the acute angles is four times as large as the other acute angle - Junior Cycle Mathematics - Question 17 - 2014

Worked Solution & Example Answer:In the right-angled triangle shown in the diagram, one of the acute angles is four times as large as the other acute angle - Junior Cycle Mathematics - Question 17 - 2014

Find the measures of the two acute angles in the triangle.

Find the slope of the line l that contains the hypotenuse of the triangle.

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