Question

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

Answer 1

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