Question

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Answer 1

Question:

In the given figure, S and T trisect the side QR of a right triangle PQR. Prove that

8PT² = 3PR² + 5PS².

Answer:

8PT² = 3PR² + 5PS²

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure, △PQR is a right-angled triangle.

We have given that, points S and T trisect the side QR.

∴ QS = ST = TR = 1 / 3 * QR

⇒ QS = ST = TR = QR / 3 - - ( 1 )

⇒ QR = 3 * ( QS = ST = TR )

⇒ QR = 3QS = 3ST = 3TR - - ( 2 )

Now, in △PQR, m∟PQR = 90°

∴ PR² = PQ² + QR² - - [ Pythagoras Theorem ]

⇒ PR² = PQ² + ( 3QS )² - - [ From ( 2 ) ]

⇒ PR² = PQ² + 9QS²

⇒ 3 * PR² = 3 * PQ² + 3 * 9QS² - - [ Multiplying each term by 3 ]

⇒ 3PR² = 3PQ² + 27QS² - - ( 3 )

Now, in △PQS, m∟PQS = 90°

∴ PS² = PQ² + QS² - - [ Pythagoras Theorem ]

⇒ PS² = PQ² + ( QR / 3 )² - - [ From ( 1 ) ]

⇒ PS² = PQ² + QR² / 9

⇒ 5 * PS² = 5 * PQ² + 5 * ( QR² / 9 ) - - [ Multiplying each term by 5 ]

⇒ 5PS² = 5PQ² + 5QR² / 9 - - ( 4 )

Now, in △PQT, m∟PQT = 90°

∴ PT² = PQ² + QT² - - [ Pythagoras Theorem ]

⇒ PT² = PQ² + ( QS + ST )² - - [ Q - S - T ]

⇒ PT² = PQ² + QS² + 2QS. ST + ST²

⇒ PT² = PQ² + ( QR / 3 )² + 2 * ( QR / 3 ) * ( QR / 3 ) + ( QR / 3 )² - - [ From ( 1 ) ]

⇒ PT² = PQ² + QR² / 9 + 2 * QR² / 9 + QR² / 9

⇒ PT² = PQ² + QR² / 9 + 2QR² / 9 + QR² / 9

⇒ PT² = PQ² + ( QR² + 2QR² + QR² ) / 9

⇒ PT² = PQ² + 4QR² / 9

⇒ 8 * PT² = 8 * PQ² + 8 * ( 4QR² / 9 ) - - [ Multiplying each term by 8 ]

⇒ 8PT² = 8PQ² + 32QR² / 9 - - ( 5 )

We have to prove that,

8PT² = 3PR² + 5PS²

⇒ ( 8PQ² + 32QR² / 9 ) = ( 3PQ² + 27QS² ) + ( 5PQ² + 5QR² / 9 ) - - [ From ( 3 ) & ( 4 ) & ( 5 ) ]

⇒ 8PQ² + 32QR² / 9 = 3PQ² + 27QS² + 5PQ² + 5QR² / 9

⇒ 8PQ² + 32QR² / 9 = 3PQ² + 5PQ² + 27QS² + 5QR² / 9

⇒ 8PQ² + 32QR² / 9 = 8PQ² + 27QS² + 5QR² / 9

⇒ 8PQ² + 32 * 3QS² / 9 = 8PQ² + 27QS² + 5 * 3QS² / 9 - - [ From ( 2 ) ]

⇒ 8PQ² + 32 * QS² / 3 = 8PQ² + 27QS² + 5 * QS² / 3

⇒ 8PQ² + 32QS² / 3 = 8PQ² + 27QS² + 5QS² / 3

⇒ 8PQ² + 32QS² = 8PQ² + 27QS² + 5QS² - - [ Cancelling 3 from both sides ]

⇒ 8PQ² + 32QS² = 8PQ² + 32QS²

∴ 8PT² = 3PR² + 5PS²

Hence proved!