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The points A, B and C lie on a circle with centre O, as shown in the diagram - HSC - SSCE Mathematics Extension 1 - Question 12 - 2017 - Paper 1
Question 12
The points A, B and C lie on a circle with centre O, as shown in the diagram. The size of ∠AOC is 100°. Find the size of ∠ABC, giving reasons.
Worked Solution & Example Answer:The points A, B and C lie on a circle with centre O, as shown in the diagram - HSC - SSCE Mathematics Extension 1 - Question 12 - 2017 - Paper 1
Step 1
Find the size of ∠ABC, giving reasons.
Answer
Given that ∠AOC = 100°, we can determine ∠ABC using the Inscribed Angle Theorem, which states that an angle formed by two points on a circle and the center of the circle is twice the angle formed at the circumference.
Therefore, we first calculate the reflex angle ∠AOB:
Next, by the Inscribed Angle Theorem:
Thus, we find that the angle ∠ABC is 130°, which is consistent with the properties of inscribed angles.
Step 2
Carefully sketch the graphs of y = |x + 1| and y = 3 – |x − 2| on the same axes, showing all intercepts.
Answer
To sketch the graphs, we first determine the critical points for each function. For the graph of y = |x + 1|, the vertex is at x = -1, and for y = 3 – |x − 2|, the vertex is at x = 2.
- For y = |x + 1|, the line increases from (-1, 0) and decreases to the left.
- For y = 3 – |x - 2|, the graph decreases to the left of x = 2 and increases thereafter.
Both graphs will intersect the x-axis and cross at specific points, producing a V-shape for the absolute value function.
Step 3
Using the graphs from part (i), or otherwise, find the range of values of x for which |x + 1| + |x − 2| = 3.
Answer
By identifying the intersection points from the sketches, we can analyze the behavior of both functions. Setting |x + 1| + |x − 2| equal to 3 allows us to find the corresponding x-values.
The valid range of x is found between the intersection points. This can be analyzed further by switching between cases based on the values of x with respect to the critical points -1 and 2.