Osmosis Tonicity Worksheet.pdf - Name: Period: Date: Worksheet ... - Free Printable

Problem Analysis:

The task involves solving a problem related to Vocabulary: Geometry of Transformations. The image contains two parts:

1. Part 1: A geometric figure (a circle) with labeled points and lines.
2. Part 2: Questions related to the geometric properties and transformations.

Let's break down the problem step by step.

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Part 1: Understanding the Figure

The figure shows:
- A circle with center \( O \).
- Points \( A \), \( B \), and \( C \) on the circumference of the circle.
- Line segments \( OA \), \( OB \), and \( OC \) are radii of the circle.
- Additional elements such as angles or other geometric relationships might be implied but are not explicitly stated in the question.

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Part 2: Solving the Questions

#### Question 1:
What is the relationship between the lengths of \( OA \), \( OB \), and \( OC \)?

- Explanation: In a circle, all radii are equal in length. Since \( OA \), \( OB \), and \( OC \) are radii of the same circle, they must have the same length.
- Answer: \( OA = OB = OC \).

#### Question 2:
If point \( D \) is the midpoint of segment \( AB \), what can you say about the line segment \( OD \)?

- Explanation:
- Point \( D \) is the midpoint of \( AB \), meaning \( AD = DB \).
- In a circle, if a line segment connects the center of the circle to the midpoint of a chord (other than the diameter), that line segment is perpendicular to the chord.
- Therefore, \( OD \) is perpendicular to \( AB \).
- Answer: \( OD \) is perpendicular to \( AB \).

#### Question 3:
If \( \angle AOB = 60^\circ \), what is the measure of \( \angle ACB \)?

- Explanation:
- \( \angle AOB \) is a central angle, and \( \angle ACB \) is an inscribed angle subtended by the same arc \( AB \).
- The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
- Given \( \angle AOB = 60^\circ \), the measure of \( \angle ACB \) is:
\[
\angle ACB = \frac{1}{2} \times \angle AOB = \frac{1}{2} \times 60^\circ = 30^\circ
\]
- Answer: \( \angle ACB = 30^\circ \).

#### Question 4:
If \( \triangle ABC \) is equilateral, what is the measure of \( \angle AOB \)?

- Explanation:
- If \( \triangle ABC \) is equilateral, then all its sides are equal, and all its internal angles are \( 60^\circ \).
- Since \( A \), \( B \), and \( C \) lie on the circumference of the circle, \( \triangle ABC \) is an inscribed triangle.
- The central angle \( \angle AOB \) subtends the same arc as the inscribed angle \( \angle ACB \). The measure of the central angle is twice the measure of the inscribed angle.
- Given \( \angle ACB = 60^\circ \) (since \( \triangle ABC \) is equilateral), the measure of \( \angle AOB \) is:
\[
\angle AOB = 2 \times \angle ACB = 2 \times 60^\circ = 120^\circ
\]
- Answer: \( \angle AOB = 120^\circ \).

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Final Answers:

1. \( OA = OB = OC \)
2. \( OD \) is perpendicular to \( AB \)
3. \( \angle ACB = 30^\circ \)
4. \( \angle AOB = 120^\circ \)

\[
\boxed{OA = OB = OC, \, OD \text{ is perpendicular to } AB, \, \angle ACB = 30^\circ, \, \angle AOB = 120^\circ}
\]