We will solve each right triangle separately. We will start with △ TUS.
△ TUS
We will find the missing measures one at a time. In this case, this means that we want to find m ∠ UTS, ST, and TU.
Angle Measure
Let's analyze △ TUS.
To find m∠ UTS, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ UTS and m ∠ S add to 90.
m ∠ UTS + m ∠ TSU = 90
Since m∠ TSU= 64, we can substitute this value in our equation and find the measure of ∠ UTS.
m ∠ UTS + 64 = 90 ⇔ m ∠ UTS = 26
Side Lengths
Let x be the length of ST. We can find its value by using the cosine ratio.
The cosine of ∠ TSU is the ratio of the length of the adjacent leg of ∠ TSU to the length of the hypotenuse.
Cosine=Adjacent/Hypotenuse ⇒ cos 64^(∘) =4/x
We can use the obtained equation to find x. We have to use a calculator to find the value of cos 64^(∘).
Therefore, the measure of ST is 9.1 m, rounded to nearest tenth.
Finally, we can find the measure of TU. To do it, we can use the Pythagorean Theorem.
a^2+b^2=c^2
For our triangle, the lengths of one of the legs and the hypotenuse are 4 and 9.1, respectively. We want to find the length of the other leg, which is TU. Let's substitute these values in the above equation.
We have found the measures of all the angles and the lengths of all the sides in △ TUS.
△ VTW
Since we already found the length of ST, we will first find the length of VT.
VT=VS+ST ⇒ VT=5+9.1=14.1
As to the rest of the missing measures, we will find them one at a time. In this case, this means that we want to find m ∠ VTW, m ∠ TVW, and TW.
Angle Measures
Let's analyze △ VTW. We can find m ∠ VTW using the sine ratio.
The sine of ∠ VTW is the ratio of the length of the leg opposite ∠ VTW to the length of the hypotenuse.
Sine=Opposite/Hypotenuse ⇒ sin ∠ VTW =9/14.1
By definition, the inverse sine of 9/14.1 is the measure of ∠ VTW. To find it, we have to use a calculator.
To find m∠ TVW, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ VTW and m ∠ TVW add to 90.
m ∠ VTW + m ∠ TVW = 90
Now, we can substitute the measure of ∠ VTW in our equation and find the measure of ∠ TVW.
39.7+ m ∠ TVW = 90 ⇔ m ∠ TVW ≈ 50.3
Side Length
Finally, we can find the length of TW. To do it, we can use the Pythagorean Theorem.
a^2+b^2=c^2
In this case, the lengths of one of the legs and the hypotenuse are 9 and 14.1, respectively. We want to find the length of the other leg, which is TW. Let's substitute these values in the above equation.
