Appendix 1: Proofs in Mathematics | A1.2

Question 6

Given that PQRSPQRS is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?

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Solution
Understand the Question
  • A quadrilateral whose diagonals bisect each other is a parallelogram, which means opposite angles are equal (∠P=∠R \angle P = \angle R and ∠Q=∠S\angle Q = \angle S).
  • In a cyclic quadrilateral, opposite angles are supplementary (∠P+∠R=180∘ \angle P + \angle R = 180^\circ).
  • Combining these properties allows us to determine the angle measures and identify the exact type of quadrilateral.

Step 1 · Use Diagonal Property to Identify Parallelogram

Since the diagonals of PQRSPQRS bisect each other, PQRSPQRS is a parallelogram.

In a parallelogram, opposite angles are equal: ∠P=∠Rand∠Q=∠S\angle P = \angle R \quad \text{and} \quad \angle Q = \angle S

Step 2 · Find the Measure of Each Angle

Diagram 1

Since PQRSPQRS is a cyclic quadrilateral, the sum of opposite angles is 180∘180^\circ: ∠P+∠R=180∘\angle P + \angle R = 180^\circ

Substituting ∠R=∠P\angle R = \angle P:

∠P+∠P=180∘2∠P=180∘∠P=180∘2=90∘\begin{aligned} \angle P + \angle P &= 180^\circ \\ 2 \angle P &= 180^\circ \\ \angle P &= \dfrac{180^\circ}{2} \\[0.6em] &= 90^\circ \end{aligned}

Since ∠P=∠R\angle P = \angle R, we have ∠R=90∘\angle R = 90^\circ.

In a parallelogram, consecutive angles sum to 180∘180^\circ:

∠P+∠Q=180∘90∘+∠Q=180∘∠Q=180∘−90∘=90∘\begin{aligned} \angle P + \angle Q &= 180^\circ \\ 90^\circ + \angle Q &= 180^\circ \\ \angle Q &= 180^\circ - 90^\circ \\ &= 90^\circ \end{aligned}

Since ∠Q=∠S\angle Q = \angle S, we have ∠S=90∘\angle S = 90^\circ.

Therefore, all four angles are 90∘90^\circ.

Step 3 · Conclude the Quadrilateral Type

A parallelogram in which all interior angles are 90∘90^\circ is a rectangle.

Therefore, PQRSPQRS is a rectangle.

Answer

The quadrilateral PQRSPQRS is a rectangle.

Common Mistakes
  • Assuming it is a Square: Concluding that PQRSPQRS is a square without information that adjacent sides are equal or diagonals are perpendicular.
  • Confusing Angle Properties: Forgetting that opposite angles are equal in a parallelogram (∠P=∠R\angle P = \angle R) but supplementary in a cyclic quadrilateral (∠P+∠R=180∘\angle P + \angle R = 180^\circ).

More questions in A1.2

Q1

Given that all women are mortal, and suppose that A is a woman, what can we conclude about A?

Q2

Given that the product of two rational numbers is rational, and suppose aa and bb are rationals, what can you conclude about abab?

Q3

Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and 17\sqrt{17} is irrational, what can we conclude about the decimal expansion of 17\sqrt{17}?

Q4

Given that y=x2+6y = x^2 + 6 and x=−1x = -1, what can we conclude about the value of yy?

Q5

Given that ABCD is a parallelogram and ∠B=80∘\angle B = 80^\circ. What can you conclude about the other angles of the parallelogram?

Q6

Given that PQRSPQRS is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?

Q7

Given that p\sqrt{p} is irrational for all primes pp and also suppose that 3721 is a prime. Can you conclude that 3721\sqrt{3721} is an irrational number? Is your conclusion correct? Why or why not?

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