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The diagram shows quadrilateral ABCD and the bisectors of the angles at A, B, C and D - HSC - SSCE Mathematics Extension 1 - Question 14 - 2018 - Paper 1
Question 14
The diagram shows quadrilateral ABCD and the bisectors of the angles at A, B, C and D. The bisectors at A and B intersect at the point P. The bisectors at A and D me... show full transcript
Worked Solution & Example Answer:The diagram shows quadrilateral ABCD and the bisectors of the angles at A, B, C and D - HSC - SSCE Mathematics Extension 1 - Question 14 - 2018 - Paper 1
Step 1
Show that PQRS is a cyclic quadrilateral
Answer
To demonstrate that quadrilateral PQRS is cyclic, we must show that the opposite angles are supplementary.
Step 1: Angle Relationships
- Let
- rac{1}{2} A = a,
- rac{1}{2} B = b,
- rac{1}{2} C = c,
- rac{1}{2} D = d.
Step 2: Sum of Opposite Angles
-
By the properties of angle bisectors and angles in a cyclic quadrilateral, we find:
- Angle PQR + Angle PSR = 180°
- (from angles at B and D)
-
Similarly,
- Angle QRS + Angle RSP = 180°
- (from angles at A and C)
Thus, since opposite angles sum to 180°, we can conclude that PQRS is a cyclic quadrilateral.
Step 2
By considering the expansions of (1 + (1 + x)^y) and (2 + y^x), show that
Answer
We start by expanding both expressions:
Expansion of (1 + (1 + x)^y)
- Using the binomial theorem, the expansion is: ext{Coefficient of } x^{r} = {y race r}
Expansion of (2 + y^x)
- Similarly, this can be expanded: ext{Coefficient of } x^{r} = {r race r}
Equating coefficients gives: {n race r} (r + 1) = {n race r}
Conclusion
This demonstrates the required relationship.
Step 3
There are 23 people who have applied to be selected for a committee of 4 people. The selection process starts with Selector A choosing a group of at least 4 people from the 23 people who applied. Selector B then chooses the 4 people to be on the committee from the group Selector A has chosen. In how many ways could this selection process be carried out?
Answer
Step 1: Choose at least 4 people
- Selector A can choose between 4 to 23 people:
Step 2: Choose 4 from the chosen group
After selector A's choice of people, Selector B has: ways to choose the committee .
Final Count
To find the total number of ways:
Evaluating the Sum
This calculation can be performed using combinatorial identities to arrive at the final total.