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What is the value of x in the diagram? 8. 30° 45

What is the value of x in the diagram? 8. 30° 45

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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What is the value of x in the diagram?

**Problem:** What is the value of \( x \) in the diagram? **Diagram Explanation:** The diagram shows a right triangle divided into two smaller right triangles by a perpendicular line from the top vertex, creating two angles of \( 30^\circ \) and \( 45^\circ \). - The larger triangle is an acute-angled triangle. - The left side of the triangle labeled as 8 is opposite the \( 30^\circ \) angle. - The \( 45^\circ \) angle is adjacent to the side labeled as \( x \). **Solution Approach:** To find the value of \( x \), use the properties of special right triangles: 1. **30-60-90 Triangle:** - The sides are in the ratio \( 1:\sqrt{3}:2 \). - If the short leg (opposite \( 30^\circ \)) is 8, then the hypotenuse is \( 8 \times 2 = 16 \), and the longer leg (adjacent) is \( 8\sqrt{3} \). 2. **45-45-90 Triangle:** - The sides are in the ratio \( 1:1:\sqrt{2} \). - Use the longer leg of the 30-60-90 triangle as the hypotenuse of the 45-45-90 triangle. - If the hypotenuse is 8 (mistake in analysis, should note corresponding leg from the above), match proportionally using correct lengths. By setting side ratios correctly, solve for \( x \). Adjust equations from recognized geometry properties and functions if needed.
Transcribed Image Text:**Problem:** What is the value of \( x \) in the diagram? **Diagram Explanation:** The diagram shows a right triangle divided into two smaller right triangles by a perpendicular line from the top vertex, creating two angles of \( 30^\circ \) and \( 45^\circ \). - The larger triangle is an acute-angled triangle. - The left side of the triangle labeled as 8 is opposite the \( 30^\circ \) angle. - The \( 45^\circ \) angle is adjacent to the side labeled as \( x \). **Solution Approach:** To find the value of \( x \), use the properties of special right triangles: 1. **30-60-90 Triangle:** - The sides are in the ratio \( 1:\sqrt{3}:2 \). - If the short leg (opposite \( 30^\circ \)) is 8, then the hypotenuse is \( 8 \times 2 = 16 \), and the longer leg (adjacent) is \( 8\sqrt{3} \). 2. **45-45-90 Triangle:** - The sides are in the ratio \( 1:1:\sqrt{2} \). - Use the longer leg of the 30-60-90 triangle as the hypotenuse of the 45-45-90 triangle. - If the hypotenuse is 8 (mistake in analysis, should note corresponding leg from the above), match proportionally using correct lengths. By setting side ratios correctly, solve for \( x \). Adjust equations from recognized geometry properties and functions if needed.
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