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3 years ago

Which must be true? Select two options. Point H is the center of the circle that passes through points D, E, and F.

Mathematics

1 answer:

olga nikolaevna [1]3 years ago

5 0

Answer:

A. Point H is the center of the circle that passes through points D, E, and F.

C. Line segment H E is-congruent-to line segment H D

Step-by-step explanation:

The center of a circle is a point within it that has the same distance to all other points on its circumference.

Constructing a circle with center H would pass through the points D, E and F at the three vertices of the given triangle. Thus producing a circumscribed circle to the triangle. A circumscribed circle is one that can is constructed outside a given figure.

It can also be observed that, DH ≅ EH ≅ FH. So that, the line segment HE is congruent to line segment HD.

The correct options in the given question are: A and C.

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Consider the matrix A. A = 1 0 1 1 0 0 0 0 0 Find the characteristic polynomial for the matrix A. (Write your answer in terms of

dusya [7]

Answer with Step-by-step explanation:

We are given that a matrix

$A=\left[\begin{array}{ccc}1&0&1\\1&0&0\\0&0&0\end{array}\right]$

a.We have to find characteristic polynomial in terms of A

We know that characteristic equation of given matrix $\mid{A-\lambda I}\mid=0$

Where I is identity matrix of the order of given matrix

I= $\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$

Substitute the values then, we get

$\begin{vmatrix}1-\lambda&0&1\\1&-\lambda&0\\0&0&-\lambda\end{vmatrix}=0$

$(1-\lambda)(\lamda^2)-0+0=0$

$\lambda^2-\lambda^3=0$

$\lambda^3-\lambda^2=0$

Hence, characteristic polynomial = $\lambda^3-\lambda^2=0$

b.We have to find the eigen value for given matrix

$\lambda^2(1-\lambda)=0$

Then , we get $\lambda=0,0,1-\lambda=0$

$\lambda=1$

Hence, real eigen values of for the matrix are 0,0 and 1.

c.Eigen space corresponding to eigen value 1 is the null space of matrix $A-I$

$E_1=N(A-I)$

$A-I=\left[\begin{array}{ccc}0&0&1\\1&-1&0\\0&0&-1\end{array}\right]$

Apply $R_1\rightarrow R_1+R_3$

$A-I=\left[\begin{array}{ccc}0&0&1\\1&-1&0\\0&0&0\end{array}\right]$

Now,(A-I)x=0[/tex]

Substitute the values then we get

$\left[\begin{array}{ccc}0&0&1\\1&-1&0\\0&0&0\end{array}\right]\left[\begin{array}{ccc}x_1\\x_2\\x_3\end{array}\right]=0$

Then , we get $x_3=0$

And $x_1-x_2=0$

$x_1=x_2$

Null space N(A-I) consist of vectors

x= $\left[\begin{array}{ccc}x_1\\x_1\\0\end{array}\right]$

For any scalar $x_1$

$x=x_1\left[\begin{array}{ccc}1\\1\\0\end{array}\right]$

$E_1=N(A-I)=Span(\left[\begin{array}{ccc}1\\1\\0\end{array}\right]$

Hence, the basis of eigen vector corresponding to eigen value 1 is given by

$\left[\begin{array}{ccc}1\\1\\0\end{array}\right]$

Eigen space corresponding to 0 eigen value

$N(A-0I)=\left[\begin{array}{ccc}1&0&1\\1&0&0\\0&0&0\end{array}\right]$

$(A-0I)x=0$

$\left[\begin{array}{ccc}1&0&1\\1&0&0\\0&0&0\end{array}\right]\left[\begin{array}{ccc}x_1\\x_2\\x_3\end{array}\right]=0$

$\left[\begin{array}{ccc}x_1+x_3\\x_1\\0\end{array}\right]=0$

Then, $x_1+x_3=0$

$x_1=0$

Substitute $x_1=0$

Then, we get $x_3=0$

Therefore, the null space consist of vectors

$x=x_2=x_2\left[\begin{array}{ccc}0\\1\\0\end{array}\right]$

Therefore, the basis of eigen space corresponding to eigen value 0 is given by

$\left[\begin{array}{ccc}0\\1\\0\end{array}\right]$

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3 years ago

James purchased a gaming computer for $2,499. He anticipates each year it will decrease in value by 7.9% from the previous year.

slava [35]

Answer:

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Step-by-step explanation:

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3 years ago

Calculate S (sub) 4 for the sequence defined by {a(sub)n}={19-4n}

Fudgin [204]

Yup, you are correct

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4 years ago

A jar contains 45 red candies and 60 black candies suppose a candy is selected at random what are the odds agianst selecting red

lara [203]

First you add up the red and black so see how many she has total.
45 + 60= 105
then if you need to find the probability of the red candies you put
45/ 105
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45+60=105
45/105= 8/21

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4 years ago

a parking garage charges $4 for the first hour and $1.50 for each additional hour. sydney has 13$ in her purse

abruzzese [7]

Answer:

7 hours

Step-by-step explanation:

8 0

2 years ago

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