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25. Rachel and Naomi are given cecle Cand point P below and are asked to construct a line tangent to circle that passes through point P Rachel draws tangentine PR and Naomi draws tangentine PN as shown What can you conclude about FN and PR Explain

25. Rachel and Naomi are given cecle Cand point P below and are asked to construct a line tangent to circle that passes through point P Rachel draws tangentine PR and Naomi draws tangentine PN as shown What can you conclude about FN and PR Explain

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Problem 25: Construction of Tangent Lines to a Circle** **Question:** Rachel and Naomi are given circle \( C \) and point \( P \) below and are asked to construct a line tangent to circle \( C \) that passes through point \( P \). Rachel draws tangent line \( \overline{PR} \) and Naomi draws tangent line \( \overline{PN} \) as shown. What can you conclude about \( \overline{PR} \) and \( \overline{PN} \)? Explain. **Diagram:** The diagram accompanying the problem shows a circle labeled \( C \) with its center at point \( O \). Outside the circle, there is a point \( P \). Two tangent lines are drawn from point \( P \) to the circle. One tangent line touches the circle at point \( R \), and the other touches the circle at point \( N \). These tangent lines are labeled \( \overline{PR} \) and \( \overline{PN} \), respectively. **Explanation and Conclusion:** For any circle, tangent lines drawn from an external point to the circle are congruent to each other. This can be concluded based on the properties of tangent lines and circles. Specifically, the lengths of \( \overline{PR} \) and \( \overline{PN} \) will be equal. ### Steps to Prove This Conclusion: 1. **Identify the Tangent Points:** In the diagram, \( R \) and \( N \) are the points of tangency. 2. **Draw the Radii to the Points of Tangency:** \( \overline{OR} \) and \( \overline{ON} \) are radii of the circle, where \( O \) is the center of the circle. 3. **Apply the Tangent Line Property:** A tangent line to a circle is perpendicular to the radius at the point of tangency. Therefore, \( \angle ORP \) and \( \angle ONP \) are both right angles. 4. **Observe the Right Triangles Formed:** Triangles \( \triangle ORP \) and \( \triangle ONP \) are right triangles. 5. **Equal Radii:** \( \overline{OR} \) is equal to \( \overline{ON} \) because they are both radii of the circle. 6. **Hypotenuse
Transcribed Image Text:**Problem 25: Construction of Tangent Lines to a Circle** **Question:** Rachel and Naomi are given circle \( C \) and point \( P \) below and are asked to construct a line tangent to circle \( C \) that passes through point \( P \). Rachel draws tangent line \( \overline{PR} \) and Naomi draws tangent line \( \overline{PN} \) as shown. What can you conclude about \( \overline{PR} \) and \( \overline{PN} \)? Explain. **Diagram:** The diagram accompanying the problem shows a circle labeled \( C \) with its center at point \( O \). Outside the circle, there is a point \( P \). Two tangent lines are drawn from point \( P \) to the circle. One tangent line touches the circle at point \( R \), and the other touches the circle at point \( N \). These tangent lines are labeled \( \overline{PR} \) and \( \overline{PN} \), respectively. **Explanation and Conclusion:** For any circle, tangent lines drawn from an external point to the circle are congruent to each other. This can be concluded based on the properties of tangent lines and circles. Specifically, the lengths of \( \overline{PR} \) and \( \overline{PN} \) will be equal. ### Steps to Prove This Conclusion: 1. **Identify the Tangent Points:** In the diagram, \( R \) and \( N \) are the points of tangency. 2. **Draw the Radii to the Points of Tangency:** \( \overline{OR} \) and \( \overline{ON} \) are radii of the circle, where \( O \) is the center of the circle. 3. **Apply the Tangent Line Property:** A tangent line to a circle is perpendicular to the radius at the point of tangency. Therefore, \( \angle ORP \) and \( \angle ONP \) are both right angles. 4. **Observe the Right Triangles Formed:** Triangles \( \triangle ORP \) and \( \triangle ONP \) are right triangles. 5. **Equal Radii:** \( \overline{OR} \) is equal to \( \overline{ON} \) because they are both radii of the circle. 6. **Hypotenuse
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