Question

In the figure PS=3,SQ=6,QR=5,PT=x and TR=y.Give any two pair of values of x and y such that line ST|| side QR

Answer 1

If PS=3, SQ=6, QR=5, PT=x and TR=y then the two pair of values of x and y such that line ST|| side QR will be (1, 2) & (2, 4).

Step-by-step explanation:

It is given that,

ST // QR

PS = 3

SQ = 6

QR = 5

PT = x

TR = y

We know that → if a line is parallel to one side of the triangle intersects the other two sides, then it divides the other two sides proportionally.  

Based on the above theorem, we get

∴  $\frac{PS}{SQ} = \frac{PT}{TR}$  

on substituting the given values

⇒  $\frac{3}{6} = \frac{x}{y}$

⇒  $\frac{1}{2} = \frac{x}{y}$

⇒  y = 2x

In Δ PQR, we have

PQ = PS + SQ = 3 + 6 = 9

and

PR = PT + TR = x + y = x + 2x = 3x

QR = 5

Also, we know that → the sum of any two sides of a triangle is greater than the 3rd side.

So, based on this, we get

PQ + QR > PR

⇒ 9 + 5 > 3x

⇒ 14 > 3x . . . . Equation 1

and

PR + QR > PQ

⇒ 3x + 5 > 9

⇒ 3x > 4 . . . . Equation 2

From equation 1 and equation 2, we get

4 < 3x < 14

Now,

The possible pair of values (considering integral values only) of x and y can be:

If x = 2 then y = 4 ⇒ value pair → (2, 4)

If x = 3 then y = 6 ⇒ value pair → (3, 6)

If x = 4 then y = 8 ⇒ value pair → (4, 8)

Thus, any two pair of values of x and y that satisfies the line ST // side QR will be (2, 4) & (3, 6).

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Answer 2

Given : PS = 3, SQ=6 QR = 5, PT =  x & TR = y

To Find : the pair of value of x &y such that ST II side QR.

​

Solution:

Thales theorem ( BPT)

if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

ST II  QR

=>  PS/SQ  =  PT / TR

=>   3 / 6  =  x/y

=> 1/2 = x/ y

=> y = 2x

PR = PT + TR  = x + 2x  = 3x

in ΔPQR

PQ = PS + SQ = 3 + 6 = 9

QR = 5

PR = 3x

Sum of two sides of a triangle is > third side

=> 5 + 3x  >  9

=> 3x  >  4

also

5 + 9 >  3x

=> 14 > 3x

=>   4 <  3x  <  14

=> x = 2  , 3  , 4    ( considering integral values only )

   y  = 4  , 6  , 8

the pair of value of x & y  = ( 2 , 4) , ( 3 , 6) and (  4, 8)

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