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Question

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 :

ABC is an isosceles triangle right-angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB . Find the ratio between the areas of ABE and ACD.


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Solution

Step 1: Note the given data and draw a diagram

Let ABC and PQR be two triangle similar triangle.

Since the triangles are similar, so

ABPQ=BCQR=ACPR …..(i)

Let AD and PS be altitudes of ABC and PQR respectively.

Step 2: Check similarity for ADB and PSQ

AA similarity: If two angles of a triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

ADB=PSQ=90°

ABD=PQSABC~PQR

According to AA similarity ADB~PSQ

So,ADPS=ABPQ ……(ii)

From (i) and (ii) we get

ABPQ=BCQR=ACPR=ADPS……(iii)

Step 3: Finding the ratio of areas of ABC and PQR

The area of a triangle is =12×Base×height

arABCarPQR=12×BC×AD12×QR×PS

arABCarPQR=BC×BCQR×QR ABPQ=BCQR=ACPR=ADPS

arABCarPQR=BC2QR2=AC2PR2=AB2PQ2

Hence proved

Step 4: Note the given data and draw a diagram that represents the given problem

Given that, ABC is an isosceles triangle right-angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB .

Let AB=BC=xcm

Pythagoras theorem: Square of the opposite side to right angle equals the sum of squares of the other two sides.

According to Pythagoras theorem,

AC2=AB2+BC2AC2=x2+x2AC2=2x2AC=2x

The diagram for the given problem is

Step 5: Find the ratio between the areas of ABE and ACD.

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 60°.

So the angles of ABE and ACD are equal to each other. According to AAA similarity ABE~ ACD.

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”

ar(ABE)ar(ACD)=ABAC2ar(ABE)ar(ACD)=x2x2ar(ABE)ar(ACD)=12ar(ABE):ar(ACD)=1:2

Hence, the ratio between the areas of ABE and ACD is 1:2.


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