Question

PQRS is a rectangle in which diagonal PR bisect ∆P as well as ∆R .show that (i) PQRS is a square. (ii) diagonal QS bisect ∆Q as well as ∆S

Answer 1

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Answer 2

Given:

• $PQRS$ is a rectangle.
• Diagonal $PR$ bisects ∠$P$ and ∠$R$.

To find:

• $PQRS$ is a square.
• Diagonal $QS$ bisects ∠$Q$ and ∠$S$.

Step by step explanation:

• We know that $PR$ bisects $P$.
• So the angles are equal.

        ∠$QPR=$∠$SPR$      ---$(1)$

• Let the above equation be $(1)$.
• We know that $PR$ bisects $R$.
• So the angle,

        ∠$QRP=$∠$SRP$      ----$(2)$

• Let the above equation be $(2)$.
• The line $PQ$ and $RS$ is parallel to each other.
• So $PR$ is a transversal,
• Thus,

        ∠$QPR=$∠$SRP$        -----$(3)$

• This is because alternate interior angles are equal.
• From equation $(1)$

        ∠$SPR=$∠$SRP$

• In the triangle Δ$PSR$,

        ∠$SPR=$∠$SRP$

• In a triangle equal angle have equal side opposite to them, So,

        $PS=SR$       -----$(4)$

• We know that the opposite side of the rectangle are equal so,

        $PQ=RS\\PS=RQ$

• Therefore from equation $(4)$ we get,

        $PQ=QR=RS=PS$

• Therefore $PQRS$ is a square.
• In the triangles Δ$PQS$ and Δ$SRQ$,
• We know that the sides,

        $PQ=RS$

        $PS=RQ$

• And,

        $SQ=SQ$  ( common in both the triangles)

• Therefore triangle Δ$PQS$ is congruent to Δ$SRQ$ by SSS congruence rule.

        Δ$PQS$≅Δ$SRQ$

• Therefore,

        ∠$PQS=$∠$RQS$   (CPCT)

• And,

        ∠$PSQ=$∠$RSQ$   (CPCT)

• Therefore, the diagonal $QS$ bisects ∠$Q$ and ∠$S$.

Final answer:

• Thus $PQRS$ is a square.
• Diagonal $QS$ bisects ∠$Q$ and ∠$S$.