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Find the value of x and y.

The main objective of this question is to find the value of $ x $ and  $ y $ in the given triangle.

This question uses the concept of a triangle. A triangle is defined by its $ 3 $ sides,  $ 3 $  angles, as well as three vertices. The total of a triangle’s internal angles will always be equal to 180 degree. This is known as a triangle’s angle sum property. The total length of any two triangle sides is bigger than that of the length of its third side.

Expert Answer

When a line splits a triangle in such a way in the line goes parallel to one of the triangle’s sides, the other sides are divided correspondingly.

Because the horizontal line stands parallel to the triangle’s base, it splits the triangle’s left as well as right sides proportionally. Thus:

\[ \space \frac{ x }{ 16 } \space = \space \frac{ y }{ 20 } \]

Now:

\[ \space \frac{ x }{ 16 } \space = \space \frac{ 45 }{ y } \]

Thus:

\[ \space \frac{ x }{ 16 } \space = \space \frac{ y }{ 20 } \] 

And:

\[ \space \frac{ x }{ 16 } \space = \space \frac{ 45}{ y } \] 

Solving for $ y $ results in:

\[ \space y^2 \space = \space 2 0( 45 ) \]

\[ \space y^2 \space = \space 900 \]

Taking the square root results in:

\[ \space y  \space = \space 3 0 \]

Now putting the value of $ y $ results in:

\[ \space \frac{ x }{ 16 } \space = \space \frac{ 30 }{ 20 } \] 

\[ \space \frac{ x }{ 16 } \space = \space \frac{ 3 }{ 2 } \] 

\[ \space x \space = \space \frac{3}{2} 16 \]

By multiplying, we get:

\[ \space x \space = \space 24 \]

Numerical Answer

The value of  $ x $ is $ 24 $, while the value of  $ y $ is  $ 30 $.

Example

How do you calculate the values of $  X  $ and $ Y $? $ Y $ seems to be the hypotenuse, $ 5 $ is indeed the neighboring side, and  $ X  $ seems to be the opposite extreme from  $ Y $, and there is a $ 30 $ degree angle in the triangle where the $ X $ and  $ Y $ lines meet.

We know that:

\[ \space \frac{1}{2} \space = \space  sin 30 \space = \space 5y \]

Now:

\[ \space \frac{1}{2} \space = \space \frac{5}{y} \]

\[ \space \frac{1 \space \times \space y}{2} \space = \space 5 \]

\[ \space y \space = \space 5 \space \times \space 2 \space = \space 10 \]

Now:

\[ \space 5^2 \space + \space x^2 \space = \space 10 \]

\[ \space x^2 \space = \space 100 \space – \space 25 \space = \space 75  \]

Solving for $ x $ results in:

\[ \space x \space = \space 5\sqrt{}3 \]

Thus the value of $ x $ is:

\[ \space x \space = \space 5\sqrt{}3 \]

And the value of  $ y $ is:

\[ \space y \space = \space 10 \]

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