To find the angles \( x \), \( y \), and \( z \) in triangle Deady (D), Fenton (F), and Grover (G), we can use the Law of Cosines. The Law of Cosines is given by:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
where \( c \) is the side opposite angle \( C \), and \( a \) and \( b \) are the other two sides.
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Calculating angle \( y \) at Deady:
Here, we denote:
- \( DF = 22 \) miles
- \( DG = 18 \) miles
- \( FG = 25 \) miles
We need to find \( y \): \[ FG^2 = DF^2 + DG^2 - 2 \cdot DF \cdot DG \cdot \cos(y) \] Plugging in the values: \[ 25^2 = 22^2 + 18^2 - 2 \cdot 22 \cdot 18 \cdot \cos(y) \] \[ 625 = 484 + 324 - 792 \cdot \cos(y) \] \[ 625 = 808 - 792 \cdot \cos(y) \] \[ 792 \cdot \cos(y) = 808 - 625 \] \[ 792 \cdot \cos(y) = 183 \] \[ \cos(y) = \frac{183}{792} \approx 0.2304 \]
Now, find \( y \): \[ y \approx \cos^{-1}(0.2304) \approx 76.6^\circ \]
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Calculating angle \( z \) at Grover:
We need to find \( z \): \[ DF^2 = FG^2 + DG^2 - 2 \cdot FG \cdot DG \cdot \cos(z) \] Plugging in the values: \[ 22^2 = 25^2 + 18^2 - 2 \cdot 25 \cdot 18 \cdot \cos(z) \] \[ 484 = 625 + 324 - 900 \cdot \cos(z) \] \[ 484 = 949 - 900 \cdot \cos(z) \] \[ 900 \cdot \cos(z) = 949 - 484 \] \[ 900 \cdot \cos(z) = 465 \] \[ \cos(z) = \frac{465}{900} \approx 0.5167 \]
Now, find \( z \): \[ z \approx \cos^{-1}(0.5167) \approx 58.3^\circ \]
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Calculating angle \( x \) at Fenton:
We know the sum of angles in a triangle is \( 180^\circ \): \[ x = 180 - y - z \] \[ x \approx 180 - 76.6 - 58.3 \approx 45.1^\circ \]
Now we can summarize the angles to the nearest degree:
- \( y \approx 77^\circ \)
- \( z \approx 58^\circ \)
- \( x \approx 45^\circ \)
Final Answers (to the nearest degree):
- The measure of \( y \) is approximately 77.
- The measure of \( z \) is approximately 58.
- The measure of \( x \) is approximately 45.
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