This image shows right triangle A B C where A is the right angle. Line segment D E intersects side A B perpendicularly at point D to create a smaller right triangle B D E. Angle B E D measures 60 degrees. Side A B measures (5 times square root 3) units. Side B C measures 'x' units.What is the value of in the figure shown?
Question
This image shows right triangle A B C where A is the right angle. Line segment D E intersects side A B perpendicularly at point D to create a smaller right triangle B D E. Angle B E D measures 60 degrees. Side A B measures (5 times square root 3) units. Side B C measures 'x' units.What is the value of in the figure shown?
Solution
To find the value of 'x' in the figure shown, we can use the properties of right triangles and trigonometry.
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We are given that angle BED measures 60 degrees. Since angle B is a right angle, angle BDE is 90 - 60 = 30 degrees.
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In right triangle BDE, we have the following relationships:
- Side BD is opposite angle BDE.
- Side DE is adjacent to angle BDE.
- Side BE is the hypotenuse.
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We can use the trigonometric ratio for tangent to find the value of side BD: tan(angle BDE) = opposite/adjacent tan(30 degrees) = BD/DE 1/sqrt(3) = BD/DE BD = DE/sqrt(3)
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We are also given that side AB measures (5 times square root 3) units. Since side AB is opposite angle B, we can equate it to side BD: AB = BD (5 times square root 3) = DE/sqrt(3)
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Solving for DE, we can multiply both sides of the equation by sqrt(3): (5 times square root 3) times sqrt(3) = DE 5 times 3 = DE DE = 15
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Now that we know the length of side DE, we can find the value of side BC using the Pythagorean theorem: BC^2 = AB^2 + AC^2 BC^2 = (5 times square root 3)^2 + 15^2 BC^2 = 75 + 225 BC^2 = 300
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Taking the square root of both sides, we find: BC = sqrt(300) BC = 10 times sqrt(3)
Therefore, the value of 'x' in the figure shown is 10 times square root 3 units.
Similar Questions
This image shows two triangles, A B C and B C D, which share side B C to form parallelogram A B D C. Side A B measures 4.3 units. Side A C measures 3 units. Side C D measures 4.3 units. Side B D measures 3 units. Angle A C B measures 61 degrees. Angle C B A measures 53 degrees. Angle C D B is labeled 'x' degrees.What is the value of in the figure?
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