Question

ABC is an isosceles triangle in which AB=AC=10cm and BC=12 cm PQRS is a rectangle inside the isosceles triangle such that P on AB and PQ =SR =y and PS =QR =x then prove that x= 6- 3y/4

Answer 1

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Answer 2

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ABC is an isosceles triangle in which AB=AC=10cm and BC=12cm. PQRS is a rectangle inside the isosceles triangle such that P on AB and PQ=SR=y and PS=QR=2x then prove that x=6-(3y/4).

Answer:

 x=6-(3y/4) is proved by properties of Isosceles triangle and rectangle.

Step-by-step explanation:

Given,  

• ABC is an isosceles triangle

            AB=AC=10cm

            BC=12cm

• PQRS is a rectangle inside the triangle on the side AB of the  triangle.

             PQ=SR= y, PS=QR=2x

• Since ABC is an Isosceles triangle,  

         ∠B and  ∠C are equal.

    from ΔABD, Tanθ = AD/BD

    from ΔACD, Tanθ = AD/CD

      now we can equate the above two angles

          AD/BD = AD/CD

           ⇒  BD = CD    

• Since BC = 12cm and  BD = CD then AD divides the line BC into two equal halves.

           So, BD = BC/2

                 BD  = 6cm

• Since AD bisects AC so AD also bisects QR.

                QD = DR= x cm

                BQ = BD-QD

                BQ = (6-x) cm

  from the triangle ABD, a right-angled triangle 90° at D.

            By Pythagoras theorem,

              AB² = AD² + BD²

               (10)² = AD² + (6)²

               100 = AD² + 36

                AD = $\sqrt{100-36}$

                 AD = 8 cm

          from ΔABD, Tanθ = AD/BD = 8/6

           from ΔPBQ, Tanθ = PQ/BQ = y/(6-x)

              By equating the above values

                   8/6 = y/(6-x)

                  4/3 =  y/(6-x)

                 4(6-x) = 3y

                     6-x = 3y/4

                        x = 6 - (3y/4)

                 Hence proved.

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