Question

A triangle PQR is drawn to circumscribe a circle of radius 8 cm such that the segments QT and TR, into which QR is divided by the point of contact T, are of lengths 14 cm and 16 cm respectively. If area of is 336 cm², find the sides PQ and PR.

Answer 1

Length of PQ =  26 cm

Length of PR =  28 cm

Given,

Area of ΔPQR = 336 cm²

we have,

ar ΔPOQ + ar ΔQOR + ar ΔPOR = 336

1/2 × PQ × OU + 1/2 × QR × OT + 1/2 × PR × OV = 336

1/2(r+14)8 + 1/2(30×8) + 1/2(r+16)8 = 336

(r+14)4 + (30×4) + (r+16)4 = 336

4r + 56 + 120 + 4r + 64 = 336

8r + 240 = 336

r = 12

Length of PQ = PU + UQ = 12 + 14 = 26 cm

Length of PR = PV + VR = 12 + 16 = 28 cm

Answer 2

The sides are PQ = 26 cm and PR = 28 cm

Explanation:

Given that PQR is a triangle and is drawn to circumscribe a circle of radius 8 cm such that the segments QT and TR, into which QR is divided by the point of contact T.

The length of QT is 14 cm and the length of TR is 16 cm

The area of the triangle is 336 cm²

We need to find the sides PQ and PR.

Since, we know that the lengths of tangents from an external point are equal.

$PU=PV=x$

$QU=QT=14$

$RV=RT=16$

And the radius of the circle is ${ OT }={OU}={OV}=8 \ {cm}$

$ar(\triangle PQR)=ar(\triangle POQ)+ar(\triangle QOR)+ar(\triangle POR)$

                  $=(\frac{1}{2} \times P Q \times O U)+(\frac{1}{2} \times Q R \times O T)+(\frac{1}{2} \times P R \times O V)$

                  $=(\frac{1}{2}(x+14) \times 8)+(\frac{1}{2} \times 30\times 8)+(\frac{1}{2} \times(x+16) \times 8)$

Simplifying, we get,

$336=(4(x+14))+(4 \times 30)+(4(x+16))$

$336=4x+56+120+4x+64$

$336=8x+240$

 $96=8x$

 $12=x$

Hence, the value of x is 12.

Therefore, the length of PQ is given by

$PQ=PU+UQ$

      $=12+14$

$PQ=26 \ cm$

The length of PR is given by

$PR=PV+VR$

      $=12+16$

$PR=28\ cm$

Hence, the sides are PQ = 26 cm and PR = 28 cm

Learn more:

(1) A triangle PQR is drawn to circumscribe a circle with radius 8cm such that segment QT and TR, into which QR is divided by the point of contact T, are of lenghts 14cm and 16cm rerespectively. If are of PQR triangle is 336sq.cm, find the sides PQ and PR.

brainly.in/question/298722

(2) A triangle Pqr is drawn to circumscribe a circle to radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lenghts 12cm and 9 cm respectively. If the area of Triangle PQR = 189cm, then find the lenghts of sides PQ and PR

brainly.in/question/1135089