Question

ABCD is a quadrilateral in which P Q R and S are midpoints of the sides ab BC CD and DA. AC is a diagonal. show that : 1),SR||AC and SR=1/2AC. 2)PQ=SR. 3)PQRS is a parallelogram.

Answer 1

Given:In quadrilateral ABCD

P,Q,R,S are mid point of sides AB,BC,CD and DA respectively

T.P: 1

2

3

In triangle ADC,

SR is a line segment joining the mid point of DA and DC respectively

:. SR||AC ( mid point therom)

In triangle BAC

PQ is a line segment joining the mid point of BA and BC respectively.

PQ||AC (mid point therom)

pQ = half of AC

From 1 &2

SR=PQ

SR ||PQ

:. PQRS is a ||gram.

Answer 2

In a  quadrilateral ABCD (1) SR || AC and SR = $\frac{1}{2}$ AC  , (2) PQ = SR  and (3) PQRS is a parallelogram is proved below.

Concept used:

Midpoint Theorem:

"We know that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of it."

Given:

In a  quadrilateral ABCD P, Q, R & S are the midpoints of the sides AB, BC, CD, and DA.  AC is diagonal.

To show:

(1) SR || AC and SR = 1/2 AC

2) PQ = SR

(3) PQRS is a parallelogram.

Proof:

Step1: Applying midpoint Theorem in △ADC :

R is the midpoint of DC and S is the midpoint of AD.

By midpoint Theorem:

SR ∥ AC ...........(i)

& SR = ½ AC ........(ii)

Part (1) SR || AC and SR = $\frac{1}{2}$ AC is proved.

Step2: Applying midpoint Theorem in △ABC :

P is the midpoint of AB and Q is the midpoint of BC.

Thus by mid point theorem,

PQ ∥ AC .......... (iii)

& PQ = $\frac{1}{2}$AC ........(iv)

Step 3: Using eq (ii) & (iv) we get:

SR = $\frac{1}{2}$ AC and PQ = $\frac{1}{2}$AC

SR = PQ = $\frac{1}{2}$AC  ...... (v)

Part (2) PQ = SR is proved.

Step 4: Using eq (i) & (iii) we get:

SR ∥ AC and PQ ∥ AC

PQ || SR ....….. (vi)

From eq.(v) and (vi) PQ = SR and PQ || SR

Since a pair of opposite sides of a quadrilateral PQRS is equal and parallel. So, PQRS is a parallelogram.

Part (3) PQRS is a parallelogram is proved.

Hence, SR || AC and SR = $\frac{1}{2}$ AC, PQ = SR, and PQRS is a parallelogram is proved.

Learn more on Brainly:

ABCD एक चतुर्भुज है जिसमें P, Q R और S क्रमशः भुजाओं AB, BC, CD और DA के मध्य-बिंदु हैं। (देखिए आकृति 8.29)I AC उसका एक विकर्ण है। दर्शाइए कि

(i) और  है।

(ii)  है।

(iii) PQRS एक समांतर चतुर्भुज है।  

brainly.in/question/10545297

ABCD एक समलंब है, जिसमें  और  है (देखिए आकृति 8.23)। दर्शाइए कि

(i)

(ii)

(iii)

(iv) विकर्ण AC=विकर्ण BD है।

[संकेत: AB को बढ़ाइए और C से होकर DA के समांतर एक रेखा खींचिए जो बढ़ी हुई भुजा AB को E पर प्रतिच्छेद करे।]

brainly.in/question/10544239

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