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Find the values of x and y for the following figure. A 30° 12 MathBits.com 60° 45⁰ X B C 60% y

Find the values of x and y for the following figure. A 30° 12 MathBits.com 60° 45⁰ X B C 60% y

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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The image presents a geometric problem that involves a composite figure made up of triangles and rectangles. Here's a detailed breakdown: The task is to find the values of \( x \) and \( y \) in the given figure. 1. **Triangle \( \triangle ABC \):** - \( \angle CAB = 30^\circ \) - \( \angle ACB = 60^\circ \) - Hypotenuse \( AB = 12 \) This is a 30-60-90 triangle. - In a 30-60-90 triangle, the ratio of the sides opposite these angles is \( 1 : \sqrt{3} : 2 \). 2. **Triangle \( \triangle BCD \):** - \( \angle BCD = 45^\circ \) - \( \angle DBC = 60^\circ \) 3. **Right Angle at Points:** - \( \angle ABC = 90^\circ \) - \( \angle BDC = 60^\circ \) 4. **Rectangle \( BCDE \):** - It shares side \( BC = x \). - The width of the rectangle is \( y \). To solve for \( x \) and \( y \): - From triangle \( \triangle ABC \), using the properties of a 30-60-90 triangle: - \( BC = x = 12 \cdot \frac{\sqrt{3}}{2} = 6\sqrt{3} \) - Using triangle \( \triangle BCD \), the height \( y \) can be calculated based on trigonometric identities or geometric properties. The angles and side lengths are critical in applying trigonometric ratios and the Pythagorean theorem to solve for \( x \) and \( y \). This problem illustrates the use of special right triangles and their properties in geometry.
Transcribed Image Text:The image presents a geometric problem that involves a composite figure made up of triangles and rectangles. Here's a detailed breakdown: The task is to find the values of \( x \) and \( y \) in the given figure. 1. **Triangle \( \triangle ABC \):** - \( \angle CAB = 30^\circ \) - \( \angle ACB = 60^\circ \) - Hypotenuse \( AB = 12 \) This is a 30-60-90 triangle. - In a 30-60-90 triangle, the ratio of the sides opposite these angles is \( 1 : \sqrt{3} : 2 \). 2. **Triangle \( \triangle BCD \):** - \( \angle BCD = 45^\circ \) - \( \angle DBC = 60^\circ \) 3. **Right Angle at Points:** - \( \angle ABC = 90^\circ \) - \( \angle BDC = 60^\circ \) 4. **Rectangle \( BCDE \):** - It shares side \( BC = x \). - The width of the rectangle is \( y \). To solve for \( x \) and \( y \): - From triangle \( \triangle ABC \), using the properties of a 30-60-90 triangle: - \( BC = x = 12 \cdot \frac{\sqrt{3}}{2} = 6\sqrt{3} \) - Using triangle \( \triangle BCD \), the height \( y \) can be calculated based on trigonometric identities or geometric properties. The angles and side lengths are critical in applying trigonometric ratios and the Pythagorean theorem to solve for \( x \) and \( y \). This problem illustrates the use of special right triangles and their properties in geometry.
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