PQRS is a parallelogram . M is the point on PS. such that PM =1÷3 QR . Prove that the quadrilateral PNRM is a parallelogram.
Given: PQRS is a parallelogram
M is a point on ps
N is point on RQ
To prove:PNRM is a parallelogram
Proof...
PS=RQ
1/3ps =1/3rq
PM =RN
PS //QR
PR//RN
HENCE PROVED....THAT.
PNRM is a parallelogram
Well, well, we have ourselves a geometry problem! Let's see if I can put a funny twist on this one!
Alright, so we have the parallelogram PQRS and a point M on the line segment PS. And we're given that PM equals one-third of QR. Our task is to prove that the quadrilateral PNRM is also a parallelogram. Challenge accepted!
Now, to prove this, we need to show that the opposite sides of PNRM are parallel. So, let's get our detective hats on and investigate.
First, since PQRS is a parallelogram, we know that opposite sides are parallel. That means, PQ is parallel to SR and PS is parallel to QR. Nothing like parallel lines to keep things running smoothly!
Now, we'll focus on the line segments QR and PM. We're told that PM is one-third of QR, which is like saying PM is the "lighter" version of QR. Maybe PM had to go on a diet to become one-third! Who knows?
Anyway, if we look closely, we can see that PM is parallel to QR. Why? Well, think about it. If PM is just a scaled-down version of QR, they must be running parallel, just like two clowns on a tightrope. They're keeping their distance!
So now, we have PM parallel to QR and PQ parallel to SR. And we know that parallel lines make the world go round. Well, at least in geometry!
Using this information, we can conclude that PN must be parallel to RS. Why? Because if you have two parallel lines and you draw a line connecting them with another line, that line becomes parallel to the other two. It's like a parallel party!
Now, we have PN parallel to RS and PM parallel to QR. And what do we call a quadrilateral with opposite sides parallel? That's right, a parallelogram!
Therefore, we can confidently state that PNRM is indeed a parallelogram. And there you have it, a geometry problem solved with a touch of clownish humor. I hope that brings a smile to your face!
To prove that quadrilateral PNRM is a parallelogram, we need to show that opposite sides are parallel.
Given: PQRS is a parallelogram, M is the point on PS such that PM = 1/3 QR.
First, let's prove that PN || MS:
Since PQRS is a parallelogram, we know that PQ || SR and PS || QR.
From this information, we can conclude that ∆PMQ and ∆SRM are similar triangles (by the AA similarity criterion) because angle PMQ = angle SRM (vertical angles) and angle PQM = angle SMR (corresponding angles).
Using the property of similar triangles, we can set up the following ratio:
PM/QM = SR/RM
Substituting the given information PM = 1/3 QR, we have:
(1/3 QR)/QM = SR/RM
Cross-multiplying, we get:
QR * RM = 3 * SR * QM
Since PQRS is a parallelogram, QR = SP, and we can substitute this in the equation:
SP * RM = 3 * SR * QM ......(1)
Next, let's prove that NR || PQ:
Using the same logic, we can show that ∆RPN and ∆QSP are similar triangles (by the AA similarity criterion) because angle RNP = angle QPS (vertical angles) and angle NPR = angle PSQ (corresponding angles).
Setting up the ratio for the similar triangles:
NR/RP = PS/SP
Since PQRS is a parallelogram, PS = QR, and we can substitute this in the equation:
NR/RP = QR/SP ......(2)
Now, let's combine equations (1) and (2):
SP * RM = 3 * SR * QM
NR/RP = QR/SP
Multiplying both sides of equation (2) by SP:
SP * NR = RP * QR
Comparing this equation to equation (1), we can see that SP * RM = SP * NR.
Since both sides of the equation are equal, we can conclude that NR || PQ.
Therefore, we have proved that PN || MS and NR || PQ, which means that quadrilateral PNRM is a parallelogram.
To prove that quadrilateral PNRM is a parallelogram, we need to show that its opposite sides are parallel.
Given:
- PQRS is a parallelogram (PQRS can be any shape)
- PM = 1/3 QR
We need to prove:
- PN || RM (opposite sides are parallel)
- NR || PM (opposite sides are parallel)
To prove that PN || RM, we can use the property of a parallelogram which states that opposite sides are parallel. By showing that QR || PM, we can use this property to conclude that PN || RM.
To prove QR || PM, we can use the property of vector addition. We can express vectors QR and PM in terms of their coordinates. Assume that Q is the origin (0,0) and let R be (a, b) (where a and b are any real numbers). Let M be (1/3a, 1/3b).
Now, we can calculate the vectors QR and PM using their coordinates:
QR = (a - 0, b - 0) = (a, b)
PM = (1/3a - 0, 1/3b - 0) = (1/3a, 1/3b)
To prove that QR || PM, we need to show that their direction vectors are proportional.
The direction vector of QR is the difference between the coordinates of R and Q: (a - 0, b - 0) = (a, b).
Similarly, the direction vector of PM is (1/3a - 0, 1/3b - 0) = (1/3a, 1/3b).
To check if the two vectors are proportional, we can compare their components. The ratio of corresponding components should be the same:
(1/3a) / a = (1/3b) / b
1 / 3 = 1 / 3
The ratio of corresponding components is equal, which means that the vectors QR and PM are proportional. Therefore, we can conclude that QR || PM.
Now, using the property of a parallelogram, we can conclude that PN || RM since both PN and RM are parallel to QR.
Hence, we have proven that quadrilateral PNRM is a parallelogram.