Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Apply the above theorem to the following :
is an isosceles triangle right-angled at . Similar triangles and are constructed on sides and . Find the ratio between the areas of and .
Step 1: Note the given data and draw a diagram
Let and be two triangle similar triangle.
Since the triangles are similar, so
…..(i)
Let and be altitudes of and respectively.
Step 2: Check similarity for and
AA similarity: If two angles of a triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
According to AA similarity
So, ……(ii)
From (i) and (ii) we get
……(iii)
Step 3: Finding the ratio of areas of and
The area of a triangle is
Hence proved
Step 4: Note the given data and draw a diagram that represents the given problem
Given that, is an isosceles triangle right-angled at . Similar triangles and are constructed on sides and .
Let
Pythagoras theorem: Square of the opposite side to right angle equals the sum of squares of the other two sides.
According to Pythagoras theorem,
The diagram for the given problem is
Step 5: Find the ratio between the areas of and .
AAA similarity: If in two triangles, the corresponding angles are equal, then the triangles are similar.
We know that each angle of an equilateral triangle is .
So the angles of and are equal to each other. According to AAA similarity .
Relationship between areas and sides of similar triangles: The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides”
Hence, the ratio between the areas of and is .