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Natalka [10]
3 years ago
15

Mhanifa please help with this I find it super difficult! Thanks! I will mark brainliest! Random answers will be reported !

Mathematics
1 answer:
Artyom0805 [142]3 years ago
4 0

Answer:

a)

The point that is equidistant to all sides of a triangle is called the <u>incenter</u>.

The incenter is located at the intersection of bisectors of the interior angles of a triangle.

b)

The point that is equidistant to all vertices of a triangle is called the <u>circumcenter</u>.  

The circumcenter is located at the intersection of perpendicular bisectors of the sides of a triangle.

c)

<em>See the attachment</em>

The blue lines and their intersection shows the incenter.

The red lines and their intersection shows the circumcenter.

As we see the red point- the <u>circumcenter </u>is closer to vertex B.

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What value of b will cause the system to have an infinite number of solutions?
irga5000 [103]

b must be equal to -6  for infinitely many solutions for system of equations y = 6x + b and -3 x+\frac{1}{2} y=-3

<u>Solution: </u>

Need to calculate value of b so that given system of equations have an infinite number of solutions

\begin{array}{l}{y=6 x+b} \\\\ {-3 x+\frac{1}{2} y=-3}\end{array}

Let us bring the equations in same form for sake of simplicity in comparison

\begin{array}{l}{y=6 x+b} \\\\ {\Rightarrow-6 x+y-b=0 \Rightarrow (1)} \\\\ {\Rightarrow-3 x+\frac{1}{2} y=-3} \\\\ {\Rightarrow -6 x+y=-6} \\\\ {\Rightarrow -6 x+y+6=0 \Rightarrow(2)}\end{array}

Now we have two equations  

\begin{array}{l}{-6 x+y-b=0\Rightarrow(1)} \\\\ {-6 x+y+6=0\Rightarrow(2)}\end{array}

Let us first see what is requirement for system of equations have an infinite number of solutions

If  a_{1} x+b_{1} y+c_{1}=0 and a_{2} x+b_{2} y+c_{2}=0 are two equation  

\Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} then the given system of equation has no infinitely many solutions.

In our case,

\begin{array}{l}{a_{1}=-6, \mathrm{b}_{1}=1 \text { and } c_{1}=-\mathrm{b}} \\\\ {a_{2}=-6, \mathrm{b}_{2}=1 \text { and } c_{2}=6} \\\\ {\frac{a_{1}}{a_{2}}=\frac{-6}{-6}=1} \\\\ {\frac{b_{1}}{b_{2}}=\frac{1}{1}=1} \\\\ {\frac{c_{1}}{c_{2}}=\frac{-b}{6}}\end{array}

 As for infinitely many solutions \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

\begin{array}{l}{\Rightarrow 1=1=\frac{-b}{6}} \\\\ {\Rightarrow6=-b} \\\\ {\Rightarrow b=-6}\end{array}

Hence b must be equal to -6 for infinitely many solutions for system of equations y = 6x + b and  -3 x+\frac{1}{2} y=-3

8 0
3 years ago
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