Prove that the area of the triangle described on one side $B C$ of a square $A B C D$ as base is half the area of the similar triangle $A C F$ produced on the diagonal $A C$ as base.

Open in App

Solution

ABCD is a square with AC as its diagonal.
Then,
$A C^{2} = A B^{2} + B C^{2} = B C^{2} + B C^{2} = 2 B C^{2}$
$\Rightarrow A C = \sqrt{2} B C$
Also, it is given that $Δ$ BCE $\sim Δ$ ACF
Using the theorem, "The area of $2$ similar triangles are in ratio of squares of the corresponding sides."
$\frac{A r e a (Δ B C F)}{A r e a (Δ A C F)} = \frac{B C^{2}}{A C^{2}} = \frac{B C^{2}}{(\sqrt{2} B C)^{2}} = \frac{1}{2}$
$\Rightarrow$ Area $(Δ B C E) = \frac{1}{2}$ .
Area $(Δ A C F)$
Hence, the area of $Δ$ BCE is half of the area of $Δ$ ACF.

Suggest Corrections

Similar questions

Q. In given figure prove that the area of the triangle BCE described on one side BC of a square ABCD as base is one half the area of the similar triangle ACF described on the diagonal AC as base.

In the adjoining figure, □ ABCD is a square . △ BCE on side BC and △ ACF on the diagonal AC are similar to each other . S how that A (△ BCE) = \frac{1}{2} A (△ ACF) .

$A B C D$ is a square. Equilateral triangles $A C F$ and $A B E$ are drawn on the the diagonal $A C$ and side $A B$ respectively. Find area of $△ A C F$ : area of $△ A B E$ .

Q. Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.
OR
if the area of two similar triangles are equal, prove that they are congruent.

Q. Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.

Join BYJU'S Learning Program

Prove that the area of the triangle described on one side BC of a square ABCD as base is half the area of the similar triangle ACF produced on the diagonal AC as base.

Prove that the area of the triangle described on one side $B C$ of a square $A B C D$ as base is half the area of the similar triangle $A C F$ produced on the diagonal $A C$ as base.