Answer

Verified

403.2k+ views

**Hint:**

In order to show that the line segment \[ER, AL\] trisect \[FC\], first join the points \[E\& L \text{ and } L\& A\]. And join the diagonal \[FC\]. Then use the property of parallelogram in order to prove \[ELAR\] is a parallelogram.

**Complete step by step solution:**

In this question, first we will modify the given diagram as given below,

As given L and R are the midpoint of EF and AC.

So the length of EL and AR is half of the length of EF and AC. The length of EL and AR is given below.

\[

EL = \dfrac{1}{2}EF \\

AR = \dfrac{1}{2}AC \\

\]

As it is given that FACE is a parallelogram. So opposite sides are parallel and equal.

Therefore \[FA = CE {\text{ and }}EL\parallel CE\]. And \[FE = CE {\text{ and }} FE\parallel AC\].

As the length of the line segment EF and AC, so it means \[EF = AC\].

So the half of the EF and AC are also equal, it is shown below.

\[\dfrac{1}{2}EF = \dfrac{1}{2}AC\].

Using the above expression for half of the EF and AC we get the following expression as given below.

\[EL = AR\].

And \[EL\parallel AR\], means EL and AR are parallel.

Thus by using the property of the parallelogram, ELAR is also a parallelogram.

So \[ER\parallel LA {\text{and }}EQ\parallel LP\]

In the triangle \[APC\] we have \[R\] is the midpoint of \[AC\], and \[P\] is the midpoint of \[FQ\].

So, the length of the line segment \[FP \& PQ\] are equal; it means \[FP = PQ \left( 1 \right)\].

In the triangle \[PFE\] we have \[L\] is the midpoint of \[FE\].

Also, as \[Q\] is the midpoint of \[CP\].

So we get the length of the line segments \[CQ \& PQ\] are equal, it means \[CQ = PQ \left( 2 \right)\]

Using the above expression (1) and (2) we find that the length of the line segments\[FP,PQ\& QC\] are equal; it means \[FP = PQ = QC\].

**Hence the above proof is showing that \[ER,AL\] trisect \[FC\].**

**Note:**

We know that in a parallelogram the opposite sides are parallel and equal. In order to prove a quadrilateral is a parallelogram it is enough to prove that one pair of opposite sides are parallel and equal.

Recently Updated Pages

How do you evaluate cos left dfrac13pi 12 right class 10 maths CBSE

How do you rewrite the inequality left 11 2x right class 10 maths CBSE

How do you solve 4 3x 025 class 10 maths CBSE

How do you find the zeros of x3 3x2 + 6x 18 class 10 maths CBSE

Consider the following statements in respect of the class 10 maths CBSE

How do you factor 2x3 + 3x2 8x 12 class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail

Name 10 Living and Non living things class 9 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths