What is true regarding two adjacent arcs created by two intersecting diameters?
A. They always have equal measures.
B. The difference of their measures is $90^{\circ}$.
C. The sum of their measures is $180^{\circ}$.
D. Their measures cannot be equal.
Answer (2)
Define the angle between the intersecting diameters as θ , so the adjacent arc measures 18 0 ∘ − θ .
Check if the arcs always have equal measures: θ = 18 0 ∘ − θ implies θ = 9 0 ∘ , which is not always true.
Check if the difference of their measures is 9 0 ∘ : ∣ θ − ( 18 0 ∘ − θ ) ∣ = 9 0 ∘ is not always true.
Check if the sum of their measures is 18 0 ∘ : θ + ( 18 0 ∘ − θ ) = 18 0 ∘ , which is always true. Therefore, the answer is: The sum of their measures is 18 0 ∘ .
Explanation
Problem Analysis Let's analyze the problem. We have two intersecting diameters in a circle, which create four arcs. We need to determine which statement is true about two adjacent arcs formed by these diameters.
Define the angles Let the angle between the two diameters be θ . Then, the measure of one arc is θ , and the measure of the adjacent arc is 18 0 ∘ − θ . We will now evaluate each of the given statements.
Evaluate each statement
They always have equal measures: This would mean θ = 18 0 ∘ − θ , which simplifies to 2 θ = 18 0 ∘ , or θ = 9 0 ∘ . This is only true when the diameters are perpendicular. Therefore, this statement is not always true.
The difference of their measures is 9 0 ∘ : This means ∣ θ − ( 18 0 ∘ − θ ) ∣ = 9 0 ∘ . Simplifying, we get ∣2 θ − 18 0 ∘ ∣ = 9 0 ∘ . This gives two possibilities: 2 θ − 18 0 ∘ = 9 0 ∘ or 2 θ − 18 0 ∘ = − 9 0 ∘ . Solving for θ , we get θ = 13 5 ∘ or θ = 4 5 ∘ . This is not always true.
The sum of their measures is 18 0 ∘ : This means θ + ( 18 0 ∘ − θ ) = 18 0 ∘ . This simplifies to 18 0 ∘ = 18 0 ∘ , which is always true.
Their measures cannot be equal: As we saw in the first statement, their measures are equal when θ = 9 0 ∘ . So, this statement is false.
Conclusion Therefore, the correct statement is: The sum of their measures is 18 0 ∘ .
Examples
Understanding arcs and diameters is crucial in many real-world applications. For example, in architecture, designing arches and curved structures requires precise calculations of arc lengths and angles. Similarly, in navigation, understanding the geometry of circles and arcs is essential for determining distances and bearings on maps and charts. Even in everyday life, knowing how arcs and diameters relate can help you estimate distances on circular objects or understand how a pizza is divided into equal slices.
The correct statement regarding two adjacent arcs created by two intersecting diameters is that the sum of their measures is always 18 0 ∘ . Therefore, the correct choice is (C).
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