Sam wants to find the height of a window in a nearby building but it is cloudy. Sam puts a mirror on the ground between himself and the building. He tilts it toward him so that when he is standing up he sees the reflection of the window. The base of the mirror is 1.22 meters from his feet and 7.32 meters from the base of the building. Sam's eye is 1.82 meters above the ground. How high up on the building is the window?
To find the height of the window, we can use similar triangles. Here's how we can do it step-by-step:
Step 1: Draw a diagram to visualize the problem. Mark the distances mentioned in the question.
Step 2: Start by identifying the similar triangles in the diagram. In this case, we have two similar triangles: one formed by Sam, the mirror, and the top of the building, and the other formed by Sam, his eye, and the reflection of the window.
Step 3: Note the corresponding sides of the similar triangles. In one triangle, the sides are: the height of the building (which we want to find), the distance from Sam's feet to the mirror (1.22 meters), and the distance from the mirror to the base of the building (7.32 meters). In the other triangle, the sides are: the height of the window (which we want to find), the distance from Sam's eye to the mirror (1.82 meters), and the distance from the mirror to Sam's feet (1.22 meters).
Step 4: Set up a proportion using the corresponding sides of the similar triangles. Since the ratios of corresponding sides in similar triangles are equal, we have:
(Height of Building) / (Distance from Sam's Feet to the Mirror) = (Height of Window) / (Distance from Sam's Eye to the Mirror)
Step 5: Substitute the values into the proportion. We have:
(Height of Building) / (1.22) = (Height of Window) / (1.82)
Step 6: Solve for the height of the window. Cross-multiply and divide:
(Height of Building) = (Height of Window) * (1.22) / (1.82)
Step 7: Substitute the known values and calculate:
(Height of Building) = (Height of Window) * 0.6703
Step 8: Since we need to find the height of the window, we can rearrange the equation:
(Height of Window) = (Height of Building) / 0.6703
Step 9: Substitute the known values and calculate:
(Height of Window) = (7.32) / 0.6703
Calculating this, we find that the height of the window is approximately 10.92 meters.
To find the height of the window, we can use similar triangles and the angle of reflection.
Step 1: Draw a diagram to visualize the problem. Label the base of the mirror as "B" and the base of the building as "C". Label Sam's eye level as "A". Also, label the top of the mirror as "D" and the top of the window as "E". Lastly, label the height of the building as "x".
D
|\
| \
x | \
| \ B
| \
| \
| \
| \
A +-------\------- C
Step 2: We know the following measurements:
- AB = 1.82 meters
- BC = 7.32 meters
- BD = 1.22 meters
Step 3: Notice that triangle ABC and triangle ADE are similar triangles since angle A is common. This means that their corresponding sides are proportional.
Step 4: Write the proportion to find the height of the window:
- AB / BC = AD / DE
Step 5: Substitute the values we know into the proportion:
- 1.82 / 7.32 = 1.22 / DE
Step 6: Cross-multiply and solve for DE:
- (1.82 * DE) = (7.32 * 1.22)
- DE = (7.32 * 1.22) / 1.82
Step 7: Calculate the height of the window:
- DE = 4.864 / 1.82 = 2.676
Step 8: Round the height of the window to an appropriate number of decimal places:
- The height of the window is approximately 2.68 meters.
Therefore, the window is approximately 2.68 meters high up on the building.