ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP=PH.
Last updated date: 15th Mar 2023
•
Total views: 206.7k
•
Views today: 4.86k
Answer
206.7k+ views
Hint: In this problem ABCD is a quadrilateral given and we have to prove that for a line GH intersecting AD, EF and BC at some specific points, then GP=PH. If we can prove that P is the midpoint of GH then the required condition will hold true. So use the property of parallelogram that opposite sides are equal and parallel to reach to the solution.
Complete step-by-step answer:
Consider ABCD parallelogram as shown above, E and F are mid-points of AB and DC (see figure) and a line GH intersects AD, EF and BC at G, P and H respectively (see figure).
Now since E and F are the midpoints of AB and DC.
$ \Rightarrow AE = EB = \dfrac{{AB}}{2}{\text{ & }}DF = FC = \dfrac{{DC}}{2}$………….. (1)
Now as we know opposite sides of a parallelogram are equal.
$ \Rightarrow AB = DC{\text{ & }}AD = BC$…………………… (2)
Therefore from equation (1) and equation (2)
$AE = EB = DF = FC = \dfrac{{AB}}{2}$
As we have proved that AE=DF and EB=FC, hence using the property that opposite sides are equal to parallelograms.
We can say that AEFD and EBCF both are parallelograms.
Now the opposite sides of a parallelogram are parallel thus using this property we can say that
$ \Rightarrow AD||EF||BC$………………….. (3) As AEFD and EBCF both are parallelograms.
Since E and F are the midpoints of AB and DC, hence P should also be the mid-point of GH by symmetry.
As P is the midpoint of GH hence
$ \Rightarrow GP = PH = \dfrac{{GH}}{2}$
Hence proved.
Note: Whenever we face such types of problems the key concept is to have the understanding of basic parallelogram properties, diagrammatic representation of the information being provided in the question always helps understanding the geometry and symmetry in the figures. This concept will help you get the right answer.
Complete step-by-step answer:

Consider ABCD parallelogram as shown above, E and F are mid-points of AB and DC (see figure) and a line GH intersects AD, EF and BC at G, P and H respectively (see figure).
Now since E and F are the midpoints of AB and DC.
$ \Rightarrow AE = EB = \dfrac{{AB}}{2}{\text{ & }}DF = FC = \dfrac{{DC}}{2}$………….. (1)
Now as we know opposite sides of a parallelogram are equal.
$ \Rightarrow AB = DC{\text{ & }}AD = BC$…………………… (2)
Therefore from equation (1) and equation (2)
$AE = EB = DF = FC = \dfrac{{AB}}{2}$
As we have proved that AE=DF and EB=FC, hence using the property that opposite sides are equal to parallelograms.
We can say that AEFD and EBCF both are parallelograms.
Now the opposite sides of a parallelogram are parallel thus using this property we can say that
$ \Rightarrow AD||EF||BC$………………….. (3) As AEFD and EBCF both are parallelograms.
Since E and F are the midpoints of AB and DC, hence P should also be the mid-point of GH by symmetry.
As P is the midpoint of GH hence
$ \Rightarrow GP = PH = \dfrac{{GH}}{2}$
Hence proved.
Note: Whenever we face such types of problems the key concept is to have the understanding of basic parallelogram properties, diagrammatic representation of the information being provided in the question always helps understanding the geometry and symmetry in the figures. This concept will help you get the right answer.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
