Question

The area of \[\Delta ABC\] is 150 square units. If \[\overline{AB}=\overline{AH}=20\] then the length of \[\overline{HG}\] is:
A) 5
B) 12
C) 16
D) 20

Answer 1

Hint: In this type of question we have to use the concept of the right angle triangle and similar triangles. We know that the area of a right angle triangle is given by \[\dfrac{1}{2}\times base\times height\]. Also we know that the right angle triangle satisfies Pythagoras theorem i.e. the square of the hypotenuse is equal to the sum of the squares of the lengths of other two sides. We know that the corresponding sides of the similar triangles are in the same ratio.

Complete step-by-step solution:
Now, we have to find the length of \[\overline{HG}\] where the area of \[\Delta ABC\] is 150 square units and \[\overline{AB}=\overline{AH}=20\].

\[\Rightarrow \text{Area of }\Delta \text{ABC = 150 sq}\text{. units}\]
\[\Rightarrow \overline{AB}=20\]
\[\Rightarrow \text{Area of }\Delta \text{ABC =}\dfrac{1}{2}\times AB\times BC\]
\[\begin{align}
  & \Rightarrow 150\text{ =}\dfrac{1}{2}\times 20\times BC \\
 & \Rightarrow 150=10\times BC \\
 & \Rightarrow \dfrac{150}{10}=BC \\
 & \Rightarrow 15\text{ units}=BC \\
\end{align}\]
Now, as \[\Delta \text{ABC}\] is right angled at \[\angle B\], by Pythagoras theorem we can write,
\[\Rightarrow A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}\]
By substituting values we get,
\[\begin{align}
  & \Rightarrow A{{C}^{2}}={{20}^{2}}+{{15}^{2}} \\
 & \Rightarrow A{{C}^{2}}=400+225 \\
 & \Rightarrow A{{C}^{2}}=625 \\
\end{align}\]
By taking square root of both sides we can write,
\[\Rightarrow AC=25\text{ units}\]
Now as \[\Delta \text{ABC}\] and \[\Delta \text{AGH}\] are similar triangles and we know that the corresponding sides of similar triangles are in the same ratio.
\[\Rightarrow \dfrac{AH}{AC}=\dfrac{HG}{CB}\] and \[\overline{AH}=20\] (By given)
\[\Rightarrow \dfrac{20}{25}=\dfrac{HG}{15}\]
On simplification we get,
\[\begin{align}
  & \Rightarrow \dfrac{20\times 15}{25}=HG \\
 & \Rightarrow HG=12\text{ units} \\
\end{align}\]
Hence, the length of \[\overline{HG}\] is 12 units.
Therefore, option B is the correct option.

Note: In this type of question students have to remember the formula for area of right angle triangle as well as Pythagoras theorem. Also students have to take care during the use of property of similar triangles as the corresponding sides are the same in ratio they have to consider corresponding sides only.