Geometric Proportions

Properties of Ratios of Areas of Two Triangles

Properties of Ratios of Areas of Two Triangles

We know how to find the area of a triangle. It is half the product of the base and the height of a triangle. Since the area of each triangle depends upon the measures of its base and height, we can find the relation between the areas of two triangles by finding the relation between their heights and bases.

Let us learn the same with the help of the following triangles.

From the figure:

$$\begin{gathered} \mathrm{Ar}\left(∆\mathrm{PQR}\right)=\frac{1}{2}\times \mathrm{Base}\times \mathrm{Height}=\frac{1}{2}\times \mathrm{QR}\times \mathrm{PS} \\ \mathrm{Ar}\left(∆\mathrm{XYZ}\right)=\frac{1}{2}\times \mathrm{Base}\times \mathrm{Height}=\frac{1}{2}\times \mathrm{YZ}\times \mathrm{XO} \end{gathered}$$

Now, let us find the ratio of the areas of ΔPQR and ΔXYZ.

Thus, the ratio of areas of two triangles is equal to the ratio of the product of their bases and corresponding heights.

Symbolically, we can write the above conclusion in the following manner:

, where A₁ and A₂ are the areas of the triangles having base and height as b₁, h₁ and b₂, h₂ respectively.

The two special cases that can arise here are given below:

(1) Triangles with same base:

If two triangles have equal or common base, then the ratio of their areas is equal to the ratio of their respective heights.

Look at the following figure.

Here, ΔABD and ΔCBD have common base, i.e. BD. Also, their respective heights are AE and CF.

∴

Symbolically:

,where A₁ and A₂ are the areas of the triangles having base and height as b, h₁ and b, h₂ respectively.

(2)Triangles with same height:

If two triangles have equal or common height, then the ratio of their areas is equal to the ratio of their respective bases.

Observe the following triangles.

Here, ΔPRS and ΔPQR have a common height, i.e. PT. Also, their respective bases are RS and QR.

∴

Symbolically:

, where A₁ and A₂ are the areas of the triangles having base and height as b₁, h and b₂, h respectively.

Let us solve some examples based on these concepts.

Example 1:

In the given figure, BD = 10 cm and BC = 3 cm.

Find the values of:

(i)

(ii)

Solution…