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Volgvan
3 years ago
5

For the triangle shown below, find the circumcenter and orthocenter

Mathematics
1 answer:
natta225 [31]3 years ago
5 0

Answer: Circumcenter = \bold{\bigg(1,-\dfrac{3}{2}\bigg)}    Orthocenter = (-4, -6)

<u>Step-by-step explanation for Circumcenter:</u>

Step 1: Find the midpoint of a line:<em> I chose (-4, 3) and (-4, -6)</em>

\bigg(\dfrac{-4-4}{2},\dfrac{3-6}{2}\bigg)=\bigg(\dfrac{-8}{2},\dfrac{-3}{2}\bigg) = \bigg(-4,-\dfrac{3}{2}\bigg)

Step 2: Find the perpendicular line that passes through that point:

Since it is a vertical line, the perpendicular line is y=-\dfrac{3}{2}

Step 3: repeat Steps 1 and 2 for another line: <em> chose (-4, -6) and (6, -6)</em>

\bigg(\dfrac{-4+6}{2},\dfrac{-6-6}{2}\bigg)=\bigg(\dfrac{2}{2},\dfrac{-12}{2}\bigg) = (1,-6)

Since it is a horizontal line, the perpendicular line is: x = 1

Step 4: Find the intersection of the two lines \bigg(y=-\dfrac{3}{2}\ \text{and}\ x = 1\bigg)

Their point of intersection is: \bigg(1, -\dfrac{3}{2}\bigg)

*************************************************************************************

<u>Step-by-step explanation for Orthocenter:</u>

Step 1: Find the perpendicular slope of a line: <em>I chose (-4, 3) and (-4, -6)</em>

Slope is undefined. Perpendicular slope is 0.

Step 2: Use the Point-Slope formula to find the equation of the line that passes through the vertex that is opposite of the line from Step 1 and has the perpendicular slope (found in Step 1).  

Vertex (6, -6) and m⊥ = 0  ⇒  y + 6 = 0(x - 6)  ⇒  y = -6

Step 3: repeat Steps 1 and 2 for another line: <em> chose (-4, -6) and (6, -6)</em>

Slope is 0. Perpendicular slope is undefined (x = __ )

Vertex (-4, 3) and m⊥ = undefined  ⇒  x = -4

Step 4: Find the intersection of the two lines (y=-6\ \text{and}\ x = -4)

Their point of intersection is: (-4, -6)

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Step-by-step explanation:

Assuming this previous info : "Trucks carry barrels of crude oil from a port in Texas to a distributer in New Mexico on long  trailers. Filling equipment is used to fill the barrels with the oil. When functioning properly,  the actual volume of oil loaded into each barrel by the filling equipment at the port is  approximately normally distributed with a mean of 55 gallons and a standard deviation of 0.5  gallons. If the mean is greater than 55.4 gallons, the filling mechanism is overfilling."

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

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Solution to the problem

Let X the random variable that represent the amount filling of a population, and for this case we know the distribution for X is given by:

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We are interested on this probability

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And the best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\sigma}

If we apply this formula to our probability we got this:

P(X>55.4)=P(\frac{X-\mu}{\sigma}>\frac{55.4-\mu}{\sigma})=P(Z>\frac{55.4-55}{0.5})=P(Z>0.8)

And we can find this probability using the complement rule:

P(Z>0.8)=1-P(Z\leq 0.8)

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

P(Z>0.8)=1-0.788=0.212

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