Answer:
The height of the tree is 6.12 meters and the height of the building is 33.12 meters.
Step-by-step explanation:
Since the person is 1.8 meters tall, HC = 1.8
And since their shadow is 10 meters long, HD = 10.
We are also given that GH is 24 meters and that FG is 150 meters.
Height of the Tree:
The height of the tree is given by GB.
Again, since m∠BGD = m∠CHD = 90°, ∠BGD ≅ ∠CHD.
Likewise, ∠D ≅ ∠D. So, by AA-Similarity:
[tex]\displaystyle \Delta BGD\sim \Delta CHD[/tex]
Corresponding parts of similar triangles are in proportion. Therefore:
[tex]\displaystyle \frac{GB}{GD}=\frac{HC}{HD}[/tex]
Note that:
[tex]GD=GH+HD[/tex]
Find GD:
[tex]GD=(24)+(10)=34[/tex]
Substitute the known values into the proportion:
[tex]\displaystyle \frac{GB}{34}=\frac{1.8}{10}[/tex]
Cross-multiply:
[tex]10GB=61.2[/tex]
Therefore:
[tex]GB=6.12\text{ meters}[/tex]
The height of the tree is 6.12 meters.
Height of the Building:
The height of the building is given by FA.
Since m∠AFD = m∠CHD = 90°, ∠AFD ≅ ∠CHD.
∠D ≅ ∠D. So, by AA-Similarity:
[tex]\Delta AFD\sim \Delta CHD[/tex]
Corresponding parts of similar triangles are in proportion. Therefore:
[tex]\displaystyle \frac{FA}{FD}=\frac{HC}{HD}[/tex]
Note that:
[tex]FD=FG+GH+HD[/tex]
Find FD:
[tex]FD=(150)+(24)+(10)=184[/tex]
Substitute the known values into the proportion:
[tex]\displaystyle \frac{FA}{184}=\frac{1.8}{10}[/tex]
Cross-multiply:
[tex]10FA=331.2[/tex]
Therefore:
[tex]FA=33.12\text{ meters}[/tex]
The height of the building is 33.12 meters.
