**Solve for \( x \). Round to the nearest tenth, if necessary.** The problem displays a right triangle \( \triangle HIJ \), where: - The angle at vertex \( I \) (labeled \( \angle HIJ \)) is \( 50^\circ \). - The angle at vertex \( J \) (labeled \( \angle HJI \)) is \( 30^\circ \). - The side opposite the right angle \( H \) and adjacent to \( I \) (labeled \( HJ \)) is the hypotenuse, and is not directly labeled with a length. - The side opposite to \( \angle HJI \) (labeled \( HI \)) is the side \( x \). To find \( x \), which is the side adjacent to the \( 30^\circ \) angle in a right triangle, you can use trigonometric ratios. Specifically, you can apply the tangent function because the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. However, to use the tangent or other trigonometric functions correctly, we typically need either an additional side's length or to recognize a special triangle (like a 30-60-90 triangle). For a 30-60-90 triangle specifically: - The side opposite the \( 30^\circ \) angle (\( JI \)) is \( \frac{1}{2} \) of the hypotenuse. - The side opposite the \( 60^\circ \) angle would be \( \frac{\sqrt{3}}{2} \) times the hypotenuse. If the triangle and values are meant to follow these special relationships exactly, it might imply assumptions or require a slightly different approach with sin, cos, or tan, given specific lengths, which are not provided directly here. Usually, with more context provided about sides, it helps directly solve using those. Make sure to refer to actual chosen trigonometrical methods or contexts for precise calculations if applicable. Otherwise, the diagram and angles given frame this standard problem understanding with trigonometry basics.
**Solve for \( x \). Round to the nearest tenth, if necessary.** The problem displays a right triangle \( \triangle HIJ \), where: - The angle at vertex \( I \) (labeled \( \angle HIJ \)) is \( 50^\circ \). - The angle at vertex \( J \) (labeled \( \angle HJI \)) is \( 30^\circ \). - The side opposite the right angle \( H \) and adjacent to \( I \) (labeled \( HJ \)) is the hypotenuse, and is not directly labeled with a length. - The side opposite to \( \angle HJI \) (labeled \( HI \)) is the side \( x \). To find \( x \), which is the side adjacent to the \( 30^\circ \) angle in a right triangle, you can use trigonometric ratios. Specifically, you can apply the tangent function because the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. However, to use the tangent or other trigonometric functions correctly, we typically need either an additional side's length or to recognize a special triangle (like a 30-60-90 triangle). For a 30-60-90 triangle specifically: - The side opposite the \( 30^\circ \) angle (\( JI \)) is \( \frac{1}{2} \) of the hypotenuse. - The side opposite the \( 60^\circ \) angle would be \( \frac{\sqrt{3}}{2} \) times the hypotenuse. If the triangle and values are meant to follow these special relationships exactly, it might imply assumptions or require a slightly different approach with sin, cos, or tan, given specific lengths, which are not provided directly here. Usually, with more context provided about sides, it helps directly solve using those. Make sure to refer to actual chosen trigonometrical methods or contexts for precise calculations if applicable. Otherwise, the diagram and angles given frame this standard problem understanding with trigonometry basics.
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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Transcribed Image Text:**Solve for \( x \). Round to the nearest tenth, if necessary.**
The problem displays a right triangle \( \triangle HIJ \), where:
- The angle at vertex \( I \) (labeled \( \angle HIJ \)) is \( 50^\circ \).
- The angle at vertex \( J \) (labeled \( \angle HJI \)) is \( 30^\circ \).
- The side opposite the right angle \( H \) and adjacent to \( I \) (labeled \( HJ \)) is the hypotenuse, and is not directly labeled with a length.
- The side opposite to \( \angle HJI \) (labeled \( HI \)) is the side \( x \).
To find \( x \), which is the side adjacent to the \( 30^\circ \) angle in a right triangle, you can use trigonometric ratios. Specifically, you can apply the tangent function because the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
However, to use the tangent or other trigonometric functions correctly, we typically need either an additional side's length or to recognize a special triangle (like a 30-60-90 triangle). For a 30-60-90 triangle specifically:
- The side opposite the \( 30^\circ \) angle (\( JI \)) is \( \frac{1}{2} \) of the hypotenuse.
- The side opposite the \( 60^\circ \) angle would be \( \frac{\sqrt{3}}{2} \) times the hypotenuse.
If the triangle and values are meant to follow these special relationships exactly, it might imply assumptions or require a slightly different approach with sin, cos, or tan, given specific lengths, which are not provided directly here. Usually, with more context provided about sides, it helps directly solve using those.
Make sure to refer to actual chosen trigonometrical methods or contexts for precise calculations if applicable. Otherwise, the diagram and angles given frame this standard problem understanding with trigonometry basics.
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