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kondaur [170]
3 years ago
10

A circle passes through points A(7,4), B(10,6), C(12,3). Show that AC must be the diameter of the circle.

Mathematics
1 answer:
Artist 52 [7]3 years ago
6 0

so we have three points, A, B and C, if indeed AC is the diameter of the circle, then half the distance of AC is its radius, and the midpoint of AC is the center of the circle, morever, since B is also on the circle, the distance from B to the center must be the same radius distance.

in short, half the distance of AC must be equals to the distance of B to the midpoint of AC, if indeed AC is the diameter.

\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{7}~,~\stackrel{y_1}{4})\qquad C(\stackrel{x_2}{12}~,~\stackrel{y_2}{3}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{12+7}{2}~~,~~\cfrac{3+4}{2} \right)\implies \left( \cfrac{19}{2}~~,~~\cfrac{7}{2} \right)=M\impliedby \textit{center of the circle}

now, let's check the distance from say A to the center, and check the distance of B to the center, if it's indeed the center, they'll be the same and thus AC its diameter.

\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ A(\stackrel{x_1}{7}~,~\stackrel{y_1}{4})\qquad M(\stackrel{x_2}{\frac{19}{2}}~,~\stackrel{y_2}{\frac{7}{2}})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ AM=\sqrt{\left( \frac{19}{2}-7 \right)^2+\left( \frac{7}{2}-4 \right)^2} \\\\\\ AM=\sqrt{\left( \frac{5}{2}\right)^2+\left( -\frac{1}{2} \right)^2}\implies \boxed{AM\approx 2.549509756796392} \\\\[-0.35em] ~\dotfill

\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ B(\stackrel{x_1}{10}~,~\stackrel{y_1}{6})\qquad M(\stackrel{x_2}{\frac{19}{2}}~,~\stackrel{y_2}{\frac{7}{2}}) \\\\\\ BM=\sqrt{\left( \frac{19}{2}-10 \right)^2+\left( \frac{7}{2}-6 \right)^2} \\\\\\ BM=\sqrt{\left( -\frac{1}{2}\right)^2+\left( -\frac{5}{2} \right)^2}\implies \boxed{BM\approx 2.549509756796392}

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pychu [463]

Answer:

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Step-by-step explanation:

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How do you write an equation of a line given y-intercept -8 and slope 3/4
Lostsunrise [7]

Answer:

y = \frac{3}{4} x-8

Step-by-step explanation:

You can us the equation y = m x + c to answer this question.

In ,

y = m x + c

m = slope

c = y - intercept

So, they have already given in the question that,

m = 3/4

c = (-8)

Therefore, you can simply put 3/4 and (-8) instead of m and c respectively like this.

y = m x + c

y = 3/4x - 8

Hope this helps you.

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4 0
2 years ago
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What is the poinet-slope form of the equation for the line in the graph?​
Ne4ueva [31]

Answer:

y -5 = \frac{9}{13}(x - 7)

Step-by-step explanation:

Given :

Two points are given in graph (-6, -4) and {7, 5).

The point-slope form of the equation of a straight line is:

y -y_{1} = m(x - x_{1})------------(1)

Let (x_{1}, y_{1})=(7,5) and (x_{2}, y_{2})=(-6,-4)

The slope of the line m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}

Put all known value in above equation.

m=\frac{-4- 5}{-6- 7}

m=\frac{-9}{-13}

m=\frac{9}{13}

The slope of the line m=\frac{9}{13}

We know m, and also know that (x_{1}, y_{1})=(7,5), so we put these value in equation 1.

y -5 = \frac{9}{13}(x - 7)

Therefore, the equation of the line is y -5 = \frac{9}{13}(x - 7).

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Attention:THIS IS PART B <em>ONLY!</em>

First,find the unit rate.

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35 divided by 2 = 17.5 so the unit rate for J.K. is 17.5 pages per hour.

now,R.L.

45/3   and  <em>x</em>/1

45 divided by 3 = 15 so the unit rate for R.L. is 15 pages per hour

now we need to divide 355 by 17.5 and divide it by 15.Whichever quotient is smaller,finishes the book first

355 divided by 17.5 = about 20 hrs----J.K.

355 divided by 15 = about 24 hrs-----R.L.

So,who read faster?


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3 years ago
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