Question
$\text{ABCD}$ is a parallelogram and $E$ is a point on $BC$ . If the diagonal $BD$ intersects $AE$ at $F$ , prove that
$A F \times F B=E F \times F D$
$A F \times F B=E F \times F D$
✓
Answer
Given: $\text{ABCD}$ is a parallelogram and $E$ is a point on $BC$ . The diagonal $BD$ intersects $AE$ at $F .$
To prove: $A F \times F B=E F \times F D$
Proof: Since ABCD is a parallelogram, then its opposite sides must be parallel.
$\therefore$ In $\triangle A D F$ and $\triangle E B F$
$\angle FDA =\angle EBF$ and $\angle F A D=\angle F E B$ [Alternate interior angles]
$\angle A F D=\angle B F E$ [vertically opposite angles]
Therefore,by $\text{AAA}$ criteria of similar triangles, we have,
$\triangle ADF=\triangle EBF$
Since the corresponding sides of similar triangles are proportional. Therefore,we have,
$\frac{A F}{F D}=\frac{E F}{F B}$
$\Rightarrow A F \times F B=E F \times F D$
To prove: $A F \times F B=E F \times F D$
Proof: Since ABCD is a parallelogram, then its opposite sides must be parallel.
$\therefore$ In $\triangle A D F$ and $\triangle E B F$
$\angle FDA =\angle EBF$ and $\angle F A D=\angle F E B$ [Alternate interior angles]
$\angle A F D=\angle B F E$ [vertically opposite angles]
Therefore,by $\text{AAA}$ criteria of similar triangles, we have,
$\triangle ADF=\triangle EBF$
Since the corresponding sides of similar triangles are proportional. Therefore,we have,
$\frac{A F}{F D}=\frac{E F}{F B}$
$\Rightarrow A F \times F B=E F \times F D$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In the given figure, from a cuboidal solid metalic block, of dimensions 15cm × 10cm × 5cm, a cylindrical hole of diameter 7cm is drilled out. Find the surface area of the remaining block. $\Big(\text{use}\ \pi=\frac{22}{7}\Big)$
→Find the coordinates of a point $A$, where $AB$ is the diameter of a circle whose centre is $(2, -3)$ and $B$ is $(1, 4).$
→State division algorithm for polynomials.
Divide $6 x^3+13 x^2+x-2$ by $2 x+1$, and find quotient and remainder.
→Write the discriminant of the following quadratic equation.
$x^2 + 2x + 4 = 0$
→$x^2 + 2x + 4 = 0$
In a family of 3 children, find the probability of having at least one boy.
Find the $n ^{\text {th }}$ term of the following APs:
$5,11,17,23, \ldots .$
→$5,11,17,23, \ldots .$
Find the value of k for which the equation $x^2 + k(2x + k - 1) + 2 = 0$ has real and equal roots.
→Using prime factorization, find the $HCF$ and $LCM$ of:
$12, 15, 21$
→$12, 15, 21$
Write the first five terms of the following sequences whose $n^{th}$ terms are:
$\text{a}_\text{n}=\frac{2\text{n}-3}{6}$
→$\text{a}_\text{n}=\frac{2\text{n}-3}{6}$