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In the figure, suppose GH = a, FG = 1, FIH is a semicircle with diameter FH, and assume Gl is perpendicular to FH. Prove: Gl = √a. Suggestion: Remember that an angle inscribed in a semicircle is a right angle (so ZFIH is a right angle). F G H

In the figure, suppose GH = a, FG = 1, FIH is a semicircle with diameter FH, and assume Gl is perpendicular to FH. Prove: Gl = √a. Suggestion: Remember that an angle inscribed in a semicircle is a right angle (so ZFIH is a right angle). F G H

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In the figure, suppose \( GH = a \), \( FG = 1 \). \( FIH \) is a semicircle with diameter \( FH \), and assume \( GI \) is perpendicular to \( FH \). Prove: \( GI = \sqrt{a} \). Suggestion: Remember that an angle inscribed in a semicircle is a right angle (so \( \angle FIH \) is a right angle). **Diagram Explanation:** The diagram shows a semicircle with the diameter \( FH \). The point \( I \) is on the circumference of the semicircle, forming the right triangle \( FIG \). The line segment \( GI \) is drawn perpendicular from point \( G \) on the diameter \( FH \) to point \( I \). - \( FG \) is marked in red and has a length of 1. - \( GH \) is marked in blue and has a length of \( a \). - Diameter \( FH \) spans the length from \( F \) to \( H \). - \( GI \) is marked in green, representing the height or perpendicular from \( G \) to the semicircle at point \( I \). This diagram and the information provided are used to prove that the length of \( GI \) is \( \sqrt{a} \). Using the properties of triangles and the Pythagorean theorem, one can calculate this.
Transcribed Image Text:In the figure, suppose \( GH = a \), \( FG = 1 \). \( FIH \) is a semicircle with diameter \( FH \), and assume \( GI \) is perpendicular to \( FH \). Prove: \( GI = \sqrt{a} \). Suggestion: Remember that an angle inscribed in a semicircle is a right angle (so \( \angle FIH \) is a right angle). **Diagram Explanation:** The diagram shows a semicircle with the diameter \( FH \). The point \( I \) is on the circumference of the semicircle, forming the right triangle \( FIG \). The line segment \( GI \) is drawn perpendicular from point \( G \) on the diameter \( FH \) to point \( I \). - \( FG \) is marked in red and has a length of 1. - \( GH \) is marked in blue and has a length of \( a \). - Diameter \( FH \) spans the length from \( F \) to \( H \). - \( GI \) is marked in green, representing the height or perpendicular from \( G \) to the semicircle at point \( I \). This diagram and the information provided are used to prove that the length of \( GI \) is \( \sqrt{a} \). Using the properties of triangles and the Pythagorean theorem, one can calculate this.
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