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Segment AB is tangent to oT at B. What is the radius of ©T? A. 25 45 T B O A. 20 о в. 25 ос. 28 O D. 10VT4

Segment AB is tangent to oT at B. What is the radius of ©T? A. 25 45 T B O A. 20 о в. 25 ос. 28 O D. 10VT4

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 Statement:** Segment \(AB\) is tangent to circle \(T\) at \(B\). What is the radius of circle \(T\)? **Diagram:** A circle labeled \(T\) with the following components: - Points \(A\), \(B\), and \(C\) are indicated around the circle. - Segment \(AB\) is tangent to the circle at \(B\). - A line segment \(TB\) is drawn from the center \(T\) to a point \(B\) on the circumference. - Another line segment \(AC\) is drawn from a point \(A\) outside the circle, passing through the circle at point \(C\) and connecting to the center point \(T\). **Given Information:** - \( \angle BAC = 25^\circ \) - \( \angle CAB = 45^\circ \) - \(\angle ACB\) is implied to be \(180^\circ - (25^\circ + 45^\circ) = 110^\circ\). **Answer Choices:** - A. 20 - B. 25 - C. 28 - D. \(10\sqrt{14}\) When explaining any diagrams: **Explanation of the Diagram:** - The diagram features a circle labeled \(T\), with a center point also labeled \(T\). - Points \(A\), \(B\), and \(C\) are marked such that \(A\) is outside the circle, \(B\) is on the circumference where the tangent touches, and \(C\) appears to be somewhere along the line extending from \(A\) through the circle. - \( \angle BAC \) and \( \angle CAB \) are provided to assist in solving for the radius of the circle. Understanding the properties of tangent lines, angles, and triangles within the circle's geometric configuration is key to solving the problem. The right-angle property of the tangent at point \(B\) with radius \(TB\), and considering the triangle properties, can guide solving the circle's radius.
Transcribed Image Text:**Problem Statement:** Segment \(AB\) is tangent to circle \(T\) at \(B\). What is the radius of circle \(T\)? **Diagram:** A circle labeled \(T\) with the following components: - Points \(A\), \(B\), and \(C\) are indicated around the circle. - Segment \(AB\) is tangent to the circle at \(B\). - A line segment \(TB\) is drawn from the center \(T\) to a point \(B\) on the circumference. - Another line segment \(AC\) is drawn from a point \(A\) outside the circle, passing through the circle at point \(C\) and connecting to the center point \(T\). **Given Information:** - \( \angle BAC = 25^\circ \) - \( \angle CAB = 45^\circ \) - \(\angle ACB\) is implied to be \(180^\circ - (25^\circ + 45^\circ) = 110^\circ\). **Answer Choices:** - A. 20 - B. 25 - C. 28 - D. \(10\sqrt{14}\) When explaining any diagrams: **Explanation of the Diagram:** - The diagram features a circle labeled \(T\), with a center point also labeled \(T\). - Points \(A\), \(B\), and \(C\) are marked such that \(A\) is outside the circle, \(B\) is on the circumference where the tangent touches, and \(C\) appears to be somewhere along the line extending from \(A\) through the circle. - \( \angle BAC \) and \( \angle CAB \) are provided to assist in solving for the radius of the circle. Understanding the properties of tangent lines, angles, and triangles within the circle's geometric configuration is key to solving the problem. The right-angle property of the tangent at point \(B\) with radius \(TB\), and considering the triangle properties, can guide solving the circle's radius.
### Question: What is cos C? ### Diagram: There is a right triangle ABC. - **Angle at A** is 90 degrees. - **AB** (opposite side to angle C) is 9 units. - **AC** (adjacent side to angle C) is \(4\sqrt{13}\) units. - **BC** (hypotenuse) is 17 units. ### Answer Choices: A. \(\frac{4\sqrt{13}}{17}\) B. \(\frac{9}{17}\) C. \(\frac{9\sqrt{13}}{52}\) D. \(\frac{4\sqrt{13}}{9}\) ### Explanation: To find the cosine of angle C, use the definition of cosine in a right triangle, which is the ratio of the adjacent side to the hypotenuse. Thus, \(\cos C = \frac{\text{adjacent}}{\text{hypotenuse}}\). Here, the side adjacent to angle C is \(4\sqrt{13}\) and the hypotenuse is 17. Therefore, \(\cos C = \frac{4\sqrt{13}}{17}\). ### Correct Answer: A. \(\frac{4\sqrt{13}}{17}\)
Transcribed Image Text:### Question: What is cos C? ### Diagram: There is a right triangle ABC. - **Angle at A** is 90 degrees. - **AB** (opposite side to angle C) is 9 units. - **AC** (adjacent side to angle C) is \(4\sqrt{13}\) units. - **BC** (hypotenuse) is 17 units. ### Answer Choices: A. \(\frac{4\sqrt{13}}{17}\) B. \(\frac{9}{17}\) C. \(\frac{9\sqrt{13}}{52}\) D. \(\frac{4\sqrt{13}}{9}\) ### Explanation: To find the cosine of angle C, use the definition of cosine in a right triangle, which is the ratio of the adjacent side to the hypotenuse. Thus, \(\cos C = \frac{\text{adjacent}}{\text{hypotenuse}}\). Here, the side adjacent to angle C is \(4\sqrt{13}\) and the hypotenuse is 17. Therefore, \(\cos C = \frac{4\sqrt{13}}{17}\). ### Correct Answer: A. \(\frac{4\sqrt{13}}{17}\)
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