In ΔXYZ, ∠X=47° and ∠Y=29°. ∠XWZ=90° and XY=3.9. Find the length of WY to the nearest 100th.

Where is W ?

Is this a typo?

To find the length of WY, we can use the properties of right triangles and trigonometry.

Step 1: Label the diagram

Let the point where WX intersects YZ be point Z.

Step 2: Find the measure of ∠Z

Since ∠XWZ is a right angle (90°), the sum of the angles in triangle XWZ is 180°.

∠XWZ + ∠W + ∠X = 180°
90° + ∠W + 47° = 180°
∠W + 137° = 180°
∠W = 180° - 137°
∠W = 43°

So, ∠W is 43°.

Step 3: Use trigonometry to find the length of WY

We can use the sine function to find the length of WY.

sin(W) = WY / WX

Substitute the known values:
sin(43°) = WY / 3.9

Rearrange the equation to solve for WY:
WY = 3.9 * sin(43°)

Using a calculator, evaluate:
WY ≈ 3.9 * 0.682

WY ≈ 2.661

Therefore, the length of WY is approximately 2.66, rounded to the nearest hundredth.

To find the length of WY, we can use the properties of right triangles and trigonometry.

Since ∠XWZ is a right angle (∠XWZ = 90°), triangle XWZ is a right triangle.

We know the measure of ∠X = 47°, ∠Y = 29°, and XY = 3.9.

To find the length of WY, we need to determine the length of WZ, and then use that information to find WY.

Let's start by finding the length of WZ using trigonometry.

Since ∠X is adjacent to WZ and ∠Y is opposite to WZ, we can use the tangent ratio to find WZ. The tangent of ∠X is defined as the ratio of the length of the side adjacent to ∠X (WZ) to the length of the side opposite of ∠X (XY).

So, tan(∠X) = WZ / XY

Plugging in the given values, we have:

tan(47°) = WZ / 3.9

Now, we can solve for WZ:

WZ = tan(47°) * 3.9

Using a calculator, we find:

WZ ≈ 4.9307

Now that we know the length of WZ is approximately 4.9307, we can use the Pythagorean theorem to find the length of WY. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (WY) is equal to the sum of the squares of the lengths of the other two sides (WZ and YZ).

So, WY^2 = WZ^2 + YZ^2

We know that the right triangle XYZ is a right triangle, so YZ, the side opposite the right angle, is the hypotenuse.

Therefore, WY^2 = WZ^2 + XY^2

Plugging in the values we found, we have:

WY^2 = (4.9307)^2 + 3.9^2

Calculating this expression using a calculator, we have:

WY^2 ≈ 24.3114 + 15.21

WY^2 ≈ 39.5214

Finally, we can take the square root of both sides to find the length of WY:

WY ≈ √39.5214

Using a calculator, we find:

WY ≈ 6.2892

Therefore, the length of WY is approximately 6.29 (rounded to the nearest hundredth).