Theorem 6.6 class 10 mathematics, theorem based on the ratio of area of two similar triangles, theorem based on the relationship between ratio of areas and the corresponding sides.
Statement :
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Given :- \[\Delta ABC\sim \Delta PQR\]
To Prove :-\[\frac{Ar(\Delta ABC)}{Ar(\Delta PQR)}=\left ( \frac{AB}{PQ} \right )^{2}=\left ( \frac{BC}{QR} \right )^{2}=\left ( \frac{AC}{PR} \right )^{2}\]
Construction :- \[Draw\: \: AM\perp BC\: \: and\: PN\perp QR\]
Proof :- \[\Delta ABC\sim \Delta PQR\]
\[\therefore \: \: \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}......(1)\]
\[and\: \: \angle A=\angle P,\: \angle B=\angle Q\: ,\: \angle C=\angle R\]
\[Area\: of\: \Delta ABC=\frac{1}{2}\times BC\times AM\]
\[Area\: of\: \Delta PQR=\frac{1}{2}\times QR\times PN\]
\[\frac{Ar(\Delta ABC)}{Ar(\Delta PQR)}=\frac{\frac{1}{2}\times BC\times AM}{\frac{1}{2}\times QR\times PN}=\frac{BC}{QR}\times \frac{AM}{PN}\]
\[Now\: in \Delta ABM and \Delta PQN\]
\[\angle B=\angle Q\: \: \: \: ......\: (\because \Delta ABC\sim \Delta PQR)\]
\[\angle 1=\angle 2\: \: ......\: \: (\because each=90^{o})\]
\[\therefore \Delta ABM\sim \Delta PQN\]
\[\Rightarrow \: \: \: \frac{AM}{PN}=\frac{AB}{PQ}\]
\[But\: \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}\]
\[\frac{AM}{PN}=\frac{BC}{QR}\]
\[\frac{Ar(\Delta ABC)}{Ar(\Delta PQR)}=\frac{BC}{QR}\times \frac{AM}{PN}=\frac{BC}{QR}\times \frac{BC}{QR}=\left ( \frac{BC}{QR} \right )^{2}...........(2)\]
From (1) and (2)
\[\frac{Ar(\Delta ABC)}{Ar(\Delta PQR)}=\left ( \frac{AB}{PQ} \right )^{2}=\left ( \frac{BC}{QR} \right )^{2}=\left ( \frac{AC}{PR} \right )^{2}\]
Hence prove the required theorem
Side-Side-Side Similarity Rule :-
If three sides of one triangle are proportional to the corresponding three sides of other triangle then triangles are similar to each other and the condition is called SSS similarity (∿) rule.
Side-Angle-Side Similarity Rule:-
If two sides of one triangle are proportional to the corresponding two sides of other triangle and the angle between the sides are equal to each other then triangles are similar to each other and the condition is called SAS similarity (∿) rule.
Angle-Angle-Angle Similarity Rule :-
If three angles of one triangle are equal to the three angles of other triangle, then triangles are similar to each other and the condition is called AAA similarity (∿) rule.
Angle-Angle Similarity Rule :-
If two angles of one triangle are equal to the two angles of other triangle, then triangles are similar to each other and the condition is called AA (∿) similarity rule.
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