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Find the values of x and y. Round to the nearest tenth.

Find the values of x and y. Round to the nearest tenth.-example-1
User Diomara
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Answer:

21) x = 22.9, y = 38.4

22) x = 14.0, y = 98.0

Explanation:

Question 21

As the interior angles of a triangle sum to 180°, the measure of the third angle of the given triangle is 100°.


\implies 180^(\circ)-52^(\circ)-28^(\circ)=100^(\circ)

As we know all interior angles and the length of one side of the triangle, we can use the Law of Sines to find the the values of x and y.


\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines (sides)} \\\\$(a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C)$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}

From inspection of the given triangle:

  • Angle 28° is opposite the side labelled x.
  • Angle 52° is opposite the side labelled y.
  • Angle 100° is opposite the side labelled 48.

Substitute these values into the Law of Sines formula:


(x)/(\sin 28^(\circ))=(y)/(\sin 52^(\circ))=(48)/(\sin 100^(\circ))

Solve for x:


\implies (x)/(\sin 28^(\circ))=(48)/(\sin 100^(\circ))


\implies x=(48\sin 28^(\circ))/(\sin 100^(\circ))


\implies x=22.882268...


\implies x=22.9\; \sf (nearest\;tenth)

Solve for y:


\implies (y)/(\sin 52^(\circ))=(48)/(\sin 100^(\circ))


\implies y=(48\sin 52^(\circ))/(\sin 100^(\circ))


\implies y=38.408020...


\implies y=38.4\; \sf(nearest\;tenth)

Therefore, the values of x and y (rounded to the nearest tenth) are:

  • x = 22.9
  • y = 38.4


\hrulefill

Question 22

As we know the lengths of two sides of the triangle and their included angle, we can use the Cosine Rule to find the measure of side x.


\boxed{\begin{minipage}{6 cm}\underline{Cosine Rule} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}

From inspection of the given triangle:

  • a = 11
  • b = 19
  • c = x
  • C = 47°

Substitute these values into the Cosine Rule and solve for x:


\implies x^2=11^2+19^2-2(11)(19)\cos 47^(\circ)


\implies x^2=482-418\cos 47^(\circ)


\implies x=\sqrt{482-418\cos 47^(\circ)}


\implies x=14.0329856...


\implies x=14.0\; \sf (nearest\;tenth)

Now use the Law of Sines to calculate angle y.

As angle y is obtuse, and the sine of an obtuse angle is the same as the sine of its supplement, then:


\implies (x)/(\sin 47^(\circ))=(19)/(\sin (180-y)^(\circ))

Rearrange the equation to isolate y:


\implies \sin (180-y)^(\circ)=(19\sin 47^(\circ))/(x)


\implies (180-y)^(\circ)=\sin^(-1)\left((19\sin 47^(\circ))/(x)\right)


\implies y^(\circ)=180^(\circ)-\sin^(-1)\left((19\sin 47^(\circ))/(x)\right)

Substitute the found value of x and evaluate:


\implies y^(\circ)=180^(\circ)-\sin^(-1)\left((19\sin 47^(\circ))/(14.0329856...)\right)


\implies y^(\circ)=180^(\circ)-81.9795708...^(\circ)


\implies y^(\circ)=98.020429...^(\circ)


\implies y=98.0\; \sf (nearest\;tenth)

Therefore, the values of x and y (rounded to the nearest tenth) are:

  • x = 14.0
  • y = 98.0
User Wootsbot
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