9303753286127016 Area Of Similar Triangles 10 Class 10

Area Of Similar Triangles 10 Class 10

 

Area Of Similar Triangles



Before learning the area of a similar triangle, let us first recall the similarity conditions. The two triangles are similar  to each other if,

  • Corresponding angles of the triangles are equal
  • Corresponding sides of  the triangles are in  proportion

If there are two triangles say ΔABC and ΔPQR, then they are similar if,

i)    ∠A=∠P, ∠B=∠Q and ∠C=∠R

ii)    ABPQ = BCQR  = ACPR

If we have two similar triangles, then not only their angles and sides share a relationship but also the ratio of their perimeter, altitudes, angle bisectors, areas and other aspects are in ratio.

In the upcoming discussion, the relation between the area of two similar triangles is discussed.

Area of Similar Triangles Theorem

Theorem: If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

To prove this theorem, consider two similar triangles ΔABC and ΔPQR;

According to the stated theorem,

area of Î”ABCarea of Î”PQR = (ABPQ)2 =(BCQR)2 = (CARP)2

As, Area of triangle = 12 × Base × Height

To find the area of ΔABC and ΔPQR, draw the altitudes AD and PE from the vertex A and P of ΔABC andΔPQR, respectively, as shown in the figure given below:






Now, area of ΔABC = 12 × BC × AD

area of ΔPQR = 12 × QR × PE

The ratio of the areas of both the triangles can now be given as:

area of Î”ABCarea of Î”PQR = 12×BC×AD12×QR×PE

⇒ area of Î”ABCarea of Î”PQR = BC × ADQR × PE ……………. (1)

Now in ∆ABD and ∆PQE, it can be seen that:

∠ABC = ∠PQR (Since ΔABC ~ ΔPQR)

∠ADB = ∠PEQ (Since both the angles are 90°)

From AA criterion of similarity ∆ADB ~ ∆PEQ

⇒ ADPE = ABPQ  …………….(2)

Since it is known that ΔABC~ ΔPQR,

ABPQ = BCQR = ACPR …………….(3)

Substituting this value in equation (1), we get

area of Î”ABCarea of Î”PQR = ABPQ × ADPE

Using equation (2), we can write



area of Î”ABCarea of Î”PQR = ABPQ × ABPQ

area of Î”ABCarea of Î”PQR =(ABPQ)2

Also from equation (3),

area of Î”ABCarea of Î”PQR = (ABPQ)2 =(BCQR)2 = (CARP)2

This proves that the ratio of the area of two similar triangles is proportional to the squares of the corresponding sides of both the triangles.

To Know how to Find the Area Of Similar Triangles, Watch The Below Video:


Balkishan Agrawal

At the helm of GMS Learning is Principal Balkishan Agrawal, a dedicated and experienced educationist. Under his able guidance, our school has flourished academically and has achieved remarkable milestones in various fields. Principal Agrawal’s vision for the school is centered on providing a nurturing environment where every student can thrive, learn, and grow.

Post a Comment

Previous Post Next Post