$\"\"$

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The right triangle is formed in the first quadrant by the $\"\"$ -axis and $\"\"$ -axis and a line through the point is $\"\"$ .

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Observe the triangle :

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The slope of $\"\"$ and $\"\"$ is equal to the slope of $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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(a)

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Find the length $\"\"$ of the hypotenuse as a function of $\"\"$ .

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Length of the hypotenuse is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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Length of the hypotenuse is $\"\"$ .

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$\"\"$

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(b)

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Graph :

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Graph the function $\"\"$ and label the minimum point.

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$\"\"$

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Observe the graph :

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The length of the hypotenuse $\"\"$ is minimum when $\"\"$ .

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$\"\"$

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(c)

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Find the vertices of the triangle such that its area is a minimum.

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The area of the right triangle is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Find the critical numbers by equating derivative to zero.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ and $\"\"$ .

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The value of $\"\"$ can not be zero.

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Therefore, $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$ .

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The vertices of the triangle are $\"\"$ and $\"\"$ .

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$\"\"$

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(a) Length of the hypotenuse is $\"\"$ .

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(b)

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Graph:

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Graph the function $\"\"$ .

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$\"\"$

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The length of the hypotenuse $\"\"$ is minimum when $\"\"$ .

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(c) The vertices of the triangle are $\"\"$ and $\"\"$ .