two trees x and y are 100m apart and y is due east of x.the bearing of a house z from x is 145degree and 225 degree. which tree is nearest to the house and by how many meters
draw the triangle XYZ
∠X = 55°
∠Y = 45°
∠X > ∠Y so YZ > XZ
∠Z = 80°
Now use the law of sines to find both XZ and YZ and their difference
YZ/sinX = XY/sinZ,
YZ/sin55 = 100/sin80,
YZ = 83.2 m.
XZ/sin45 = XY/sin80,
XZ/sin45 = 100/sin80,
XZ = ?
lamara muhammad
To determine which tree is nearest to the house, we can use trigonometry to find the distances between the house and each tree.
Let's consider the bearing of the house from Tree X, which is 145 degrees.
1. Draw a diagram and label the points. Place Tree X at the origin (0,0), Tree Y at (100,0), and the house Z at an unknown point.
2. Since the bearing is measured clockwise from the north direction, we can visualize the bearing of 145 degrees as a line starting from Tree X and forming a 35-degree angle with the positive x-axis.
3. Now, using trigonometry, we can find the coordinates of the house in terms of x and y.
- The x-coordinate of House Z can be found by using the cosine of the angle: x-coordinate = x + distance * cos(angle).
- Similarly, the y-coordinate of House Z can be found by using the sine of the angle: y-coordinate = y + distance * sin(angle).
4. Since the bearing of the house is given as 145 degrees, we can fill in the values: x-coordinate = 0 + distance * cos(145) and y-coordinate = 0 + distance * sin(145).
5. To find the distance between Tree X and House Z, we can use the distance formula: distance = sqrt((x-coordinate)^2 + (y-coordinate)^2).
6. Calculate the distance between Tree X and House Z using the above formula.
7. Repeat this process for the bearing of 225 degrees from Tree X to find the distance between Tree Y and House Z.
8. Compare the distances obtained in steps 6 and 7. The tree that is nearest to the house is the one corresponding to the shorter distance.
By following these steps, you should be able to determine which tree is nearest to the house and by how many meters.