Final 2 - Height of the Church

Given:

• $\mbox{Observer #1 is at ground level} $
• $\mbox{Sees the top of  head of Observer #2 and top of church in line.}   $
• $\mbox{Observer #2} = 1 \; \mbox{meter tall}   $
• $\mbox{Meters from Observer #1} = 2   $
• $ \mbox{Meters from church} = 90  $

Question:

• $\mbox{How tall is the church?} $

Rationale:

• $\mbox{We have two similar triangles one within the other.} $
• $ \mbox{A small one: 1 meters vertical by  2 meters horizontal describing an angle} = \theta   $
• $ \mbox{A large one: the height of church vertical by 92 meters horizontal  describing an angle} = \theta $
• $ \dfrac{vertical_{small}}{horizontal_{small}} = \tan\theta   $
• $ \dfrac{vertical_{church}}{horizontal_{church}} = \tan\theta $

Calculate:

• $\tan\theta = \dfrac{1}{2} \; \mbox{using the small triangle} $
• $ \dfrac{vertical_{church}}{horizontal_{church}} = \tan\theta $
• $\dfrac{vertical_{church}}{92} = \tan\theta = 0.5   $
• $ vertical_{church} = 0.5 * 92 = 46 \; \mbox{meters}   $

References:

• Similar Triangles