In triangle , side has length inches, angle is , and angle is . Find the length of side to the nearest inch.
Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
- OLD 1. Angles and the Trigonometric Functions Coming soon
- OLD 2. Trigonometric Functions graphs, Inverse Trigonometric Functions Coming soon
- OLD 3. Trigonometric Identities and Equations Coming soon
- OLD 4. Laws of Sines, Cosines and Vectors Coming soon
- OLD 5. Complex Numbers, Polar Coordinates and Parametric Equations Coming soon
- NEW (not used) 7. Laws of Sines, Cosines and Vectors Coming soon
- NEW (not used) 8. Vectors Coming soon
- NEW(not used) 9. Polar equations Coming soon
- NEW (not used) 11. Graphing Complex Numbers Coming soon
7. Non-Right Triangles
Law of Sines
Multiple Choice
In triangle , an angle bisector from vertex is perpendicular to side . If and , what is the length of ?
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Verified step by step guidance1
Identify the given elements: In triangle FGH, the angle bisector from vertex G is perpendicular to side FH. The lengths FG = 8 and GH = 16 are given, and we need to find the length of FH.
Recall the Angle Bisector Theorem, which states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. If the angle bisector from G meets FH at point I, then \( \frac{FI}{IH} = \frac{FG}{GH} = \frac{8}{16} = \frac{1}{2} \).
Since the angle bisector is perpendicular to FH, triangle GIH is a right triangle with GI as the angle bisector and GI \(\perp\) FH. This means GI is the height from G to FH.
Let the length of FH be \( x \). Using the ratio from the angle bisector theorem, we can express the segments as \( FI = \frac{x}{3} \) and \( IH = \frac{2x}{3} \).
Apply the Pythagorean theorem to triangles FGI and GHI, using the known lengths FG = 8 and GH = 16, and the segments FI and IH expressed in terms of \( x \). Set up equations and solve for \( x \), the length of FH.
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