Question

In the given figure, the cross section of a roof of a house is shown.

I.Identify and write the names of similar triangles.
II.By which theorem, triangles are similar?
III.Find the height h of the roof.

Answer 1

Hint: In this given problem, observe the given diagram carefully and decide which theorem can be used to find out the similar triangles. Here, we will use the Pythagorean Theorem first and then with the help of similar triangles, will take ratios of the triangle to find the unknown height. For the resultant value, place the given values and simplify.

Complete step-by-step answer:
By the Pythagorean Theorem - If an altitude is drawn on hypotenuse of the right triangle, then triangles formed are similar to the original triangle to one another.
I. $ \Rightarrow \Delta ACB \sim \Delta DAB \sim \Delta DCA $

II.) By using the Pythagorean Theorem, we get three triangles similar to each other.

III.) Now, referring to the given diagram
by using the fact that –
 $ \Delta CAD \sim \Delta CBA $
The sides of the similar triangles are in proportion –
 $ \dfrac{{AD}}{{BA}} = \dfrac{{CA}}{{CB}} $
Place the values in the above equation –
 $ \dfrac{h}{8} = \dfrac{6}{{10}} $
Simplify and make the unknown height “h” the subject –
Cross-multiplication implies
 $
  h = \dfrac{{6 \times 8}}{{10}} \\
  h = \dfrac{{48}}{{10}} \\
  h = 4.8m \;
  $
Hence, the height of the roof is $ 4.8\;m $
So, the correct answer is “4.8 m”.

Note: Remember the difference between all the three theorem statements for the Pythagorean theorems and the geometric mean theorem and its applications. Once, the proper and correct ratios of sides are formed, rest simplification goes well in these types of problems.