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3 years ago

14

For the following diagram, assume that

1 answer:

padilas [110]3 years ago

3 0

Answer:

x = 80

y = 15

Step-by-step explanation:

The angle 6y = 90 degrees, because it forms a straight angle with the right angle next to it. 6y and the right angle next to it are also called supplementary angles (added up = 180 degrees)

6y = 90

y =15

x and the angle underneath it are also supplementary angles. We know the angle under x = 100 degrees, because that angle and the one marked 100 degrees are corresponding angles.

x + 100 =180

x = 80

Another way to solve for x is to look at the quadrilateral formed by the lines P and Q with L and M. The interior angles of a quadrilateral add up to 360 degrees. We know 6y = 90 degrees, and the angle above is is also 90, because L&M are parallel and P is perpendicular to them.

360 - 90 - 90 - 100 = 80

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Find below the calculations of the two areas, each with two methods. The results are:

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$Area=5000\sqrt{3}units^2$

Lower triangle:

$Area=14,530m^2$

Explanation:

<u>A) Method 1</u>

When you are not given the height, but you are given two sides and the included angle between the two sides, you can use this formula:

$Area=side_1\times side_2\times sin(\alpha)$

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1. <u>Upper triangle:</u>

$side_1=200units\\ \\ side_2=100units\\ \\ \alpha =60\º\\ \\ Area=200units\times 100units\times sin(60\º)/2\\ \\ Area=5000\sqrt{3}units^2$

2. <u>Lower triangle:</u>

$side_1=231m\\ \\ side_2=150m\\ \\ \alpha =123\º\\ \\ Area=231m\times 150m\times sin(123\º)/2\\ \\ Area=14,529.96m^2\approx14,530m^2$

<u></u>

<u>B) Method 2</u>

You can find the height of the triangle using trigonometric properties, and then use the very well known formula:

$Area=(1/2)\times base\times height$

Use it for both triangles.

3. <u>Upper triangle:</u>

The trigonometric ratio that you can use is:

$sine(\alpha)=opposite\text{ }leg/hypotenuse$

Notice the height is the opposite leg to the angle of 60º, and the side that measures 100 units is the hypotenuse of that right triangle. Then:

$sin(60\º)=height/100units\\ \\ height=sin(60\º)\times100units\\ \\ height=50\sqrt{3}units$

$Area=(1/2)\times base\times height=(1/2)\times 200units\times 50\sqrt{3}units=5,000\sqrt{3}units^2$

3. <u>Lower triangle:</u>

<u />

$sin(180\º-123\º)=height/231m\\ \\ height=sin(57\º)\times 231m\\ \\ height=193.7329m^2$ <u />

<u />

<u /> $Area=(1/2)\times base\times height=(1/2)\times 150m\times 193.7329m^2\\\\ Area=14,529.96m^2\approx 14,530m^2$ <u />

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3 years ago