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# Coordinates

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A point (3*sqrt(5), d + 6) is 3d units away from the origin. What is the smallest possible value of d?

Jan 21, 2022

### 1+0 Answers

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To graph it out, we would see a right triangle with legs of length $$3\sqrt{5}$$ and $$d + 6$$. If the hypotenuse is $$3d$$, then we can use the pythagorean theorem to get this equation:

$$(3\sqrt{5})^2 + (d + 6)^2 = (3d)^2$$

Simplified:

$$45 + d^2 + 36 + 12d = 9d^2$$

Combine like terms and subtraction:

$$-8d^2 + 12d + 81 = 0$$

Now we have a quadratic, we can apply the quadratic formula $$d$$ = $$-b {+\over} \sqrt{b^2 - 4ac}\over2a$$ where a is the coefficient of $$d^2$$, b is the coefficient of $$d$$, and c is the constant of the equation.

Plugging in the values, we get:

$$-3 {+\over} 3\sqrt{19}\over-4$$ = d

Since we need the smallest value of d, and d can't be a negative distance away from something, then we will use the subtraction operation. This simplifies to:

$$d = {3 + 3\sqrt{19}\over4}$$

Jan 23, 2022