Thediagonals of a parallelogram `A B C D`intersectat `Odot`A linethrou – InterviewSolution

InterviewSolution

1.	Thediagonals of a parallelogram `A B C D`intersectat `Odot`A linethrough `O`meets `A B `in `x a n d C D`in `Ydot`Show that`a r (A X Y X)=1/2(a r\|""\|^(gm) A B C D)`
Answer» Diagonal AC of \|\|gm ABCD divides it into two triangles of equal area. `therefore ar(squareACD)=(1)/(2)ar("\|\|"gm ABCD)" "...(i)` In `triangle` OAP and OCQ, we have: OA = OC (diagonals of a \|\|gm bisect each other) `angleAOP=angleCOQ " "("vert. opp." angle)` `angle PAO=angleQCO " " ("alt. interior " angle)` `therefore triangle OAPcongtriangleOCQ.` `therefore ar(triangleOAP)=ar(triangleOCQ)` `rArr ar(triangleOAP)+ar("quad. AOQD")` `=ar(triangleOCQ)+ar("quad. AOQD")` `rArr ar("quad. APQD")=ar(triangleACD)` `=(1)/(2)ar("\|\|gm ABCD") " " ["using (i)"]`. `therefore ar(squareAPQD)=(1)/(2)ar("\|\|gm ABCD")`.

Discussion

No Comment Found

Related InterviewSolutions

In a parallelogram ABCD, it is being given that AB = 10 cm and the altitudes corresponding to the sides AB and AD are DL = 6 cm and BM = 8 cm, respectively. Find AD.
In Fig. 9.31, ABCD, DCFE and ABFE are parallelograms. Show that `a r (A D E) = a r(B C F)dot`
Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that`a r (A P B) xx a r (C P D) = a r (A P D) xx a r (B P C)dot`
In a triangle ABC, E is the mid-point of median AD. Show that `a r (B E D)=1/4a r (A B C)```
XY is a line parallel to side BC of a triangle ABC. If `B E || A C`and `C F || A B`meet XY at E and F respectively, show that `a r (A B E) = a r (A C F)`
In the adjoining figure, ABCD is a parallelogram and P is any points on BC. Prove that `ar(triangleABP)+ar(triangleDPC)=ar(trianglePDA)`.
In figure, ABCD and AEFD are two parallelograms. Prove that `ar (DeltaPEA) = ar (DeltaQFD)`.
In figure, ABCD and AEFD are two parallelograms. Prove that `ar (DeltaPEA) = ar (DeltaQFD)`.
In an equilateral `Delta ABC, D ` is the midpoint of AB and E is the midpoint of AC. Then, `ar (Delta ABC) : ar (Delta ADE)=?` A. `1 : 2`B. `1 : 4`C. `sqrt3 : 2`D. `3 : 4`
A farmer was having a field in the form of a parallelogram PQRS. She took any point Aon RS and joined it to points P and Q. In how many parts the fields is divided? Whatare the shapes of these parts? The farmer wants to sow wheat and pulses in equalportions of the field separately. How should she do it?