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bazaltina [42]
2 years ago
5

The coordinates of the three vertices of the triangle are A (x1, y1), B (x2, y2), C (x3, y3).

Mathematics
1 answer:
kaheart [24]2 years ago
6 0

Answer:

Step-by-step explanation:

Using the section formula, if a point (x,y) divides the line joining the points (x  

1

​

,y  

1

​

) and (x  

2

​

,y  

2

​

) in the ratio m:n, then  

(x,y)=(  

m+n

mx  

2

​

+nx  

1

​

 

​

,  

m+n

my  

2

​

+ny  

1

​

 

​

)

The vertices of the triangle are given to be (x  

1

​

,y  

1

​

),(x  

2

​

,y  

2

​

) and (x  

3

​

,y  

3

​

). Let these vertices be A,B and C respectively.  

Then the coordinates of the point P that divides AB in l:k will be

(  

l+k

lx  

2

​

+kx  

1

​

 

​

,  

l+k

ly  

2

​

+ky  

1

​

 

​

)

The coordinates of point which divides PC in m:k+l will be  

⎩

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎧

​

 

m+k+l

mx  

3

​

+(k+l)  

(l+k)

lx  

2

​

+kx  

1

​

 

​

 

​

,  

m+k+l

my  

3

​

+(k+l)  

(l+k)

ly  

2

​

+ky  

1

​

 

​

 

​

 

⎭

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎫

​

 

⇒(  

m+k+l

kx  

1

​

+lx  

2

​

+mx  

3

​

 

​

,  

m+k+l

ky  

1

​

+ly  

2

​

+my  

3

​

 

​

)

You might be interested in
a) What is an alternating series? An alternating series is a whose terms are__________ . (b) Under what conditions does an alter
andriy [413]

Answer:

a) An alternating series is a whose terms are alternately positive and negative

b) An alternating series \sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} (-1)^{n-1} b_n where bn = |an|, converges if 0< b_{n+1} \leq b_n for all n, and \lim_{n \to \infty} b_n = 0

c) The error involved in using the partial sum sn as an approximation to the total sum s is the remainder Rn = s − sn and the size of the error is bn + 1

Step-by-step explanation:

<em>Part a</em>

An Alternating series is an infinite series given on these three possible general forms given by:

\sum_{n=0}^{\infty} (-1)^{n} b_n

\sum_{n=0}^{\infty} (-1)^{n+1} b_n

\sum_{n=0}^{\infty} (-1)^{n-1} b_n

For all a_n >0, \forall n

The initial counter can be n=0 or n =1. Based on the pattern of the series the signs of the general terms alternately positive and negative.

<em>Part b</em>

An alternating series \sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} (-1)^{n-1} b_n where bn = |an|  converges if 0< b_{n+1} \leq b_n for all n and \lim_{n \to \infty} b_n =0

Is necessary that limit when n tends to infinity for the nth term of bn converges to 0, because this is one of two conditions in order to an alternate series converges, the two conditions are given by the following theorem:

<em>Theorem (Alternating series test)</em>

If a sequence of positive terms {bn} is monotonically decreasing and

<em>\lim_{n \to \infty} b_n = 0<em>, then the alternating series \sum (-1)^{n-1} b_n converges if:</em></em>

<em>i) 0 \leq b_{n+1} \leq b_n \forall n</em>

<em>ii) \lim_{n \to \infty} b_n = 0</em>

then <em>\sum_{n=1}^{\infty}(-1)^{n-1} b_n  converges</em>

<em>Proof</em>

For this proof we just need to consider the sum for a subsequence of even partial sums. We will see that the subsequence is monotonically increasing. And by the monotonic sequence theorem the limit for this subsquence when we approach to infinity is a defined term, let's say, s. So then the we have a bound and then

|s_n -s| < \epsilon for all n, and that implies that the series converges to a value, s.

And this complete the proof.

<em>Part c</em>

An important term is the partial sum of a series and that is defined as the sum of the first n terms in the series

By definition the Remainder of a Series is The difference between the nth partial sum and the sum of a series, on this form:

Rn = s - sn

Where s_n represent the partial sum for the series and s the total for the sum.

Is important to notice that the size of the error is at most b_{n+1} by the following theorem:

<em>Theorem (Alternating series sum estimation)</em>

<em>If  \sum (-1)^{n-1} b_n  is the sum of an alternating series that satisfies</em>

<em>i) 0 \leq b_{n+1} \leq b_n \forall n</em>

<em>ii) \lim_{n \to \infty} b_n = 0</em>

Then then \mid s - s_n \mid \leq b_{n+1}

<em>Proof</em>

In the proof of the alternating series test, and we analyze the subsequence, s we will notice that are monotonically decreasing. So then based on this the sequence of partial sums sn oscillates around s so that the sum s always lies between any  two consecutive partial sums sn and sn+1.

\mid{s -s_n} \mid \leq \mid{s_{n+1} -s_n}\mid = b_{n+1}

And this complete the proof.

5 0
3 years ago
Can you guys help and put these in order?
PtichkaEL [24]
-3, -2.5, -3/4, 1/4, 3.1, 16/5, 6. I can’t see the other numbers, so it’s just these
7 0
2 years ago
Alex made 135 out of 540 cookies for the bake sale. What percent of cookies did Alex make for the bake sale?
dsp73

Answer:

c) 25%

Step-by-step explanation:

135÷540=0.25

so 25%

4 0
2 years ago
PLEASE help me with this. A parking garage has floors above and below ground level. For a scavenger hunt, Gaia's friends are giv
almond37 [142]

You can use -4, -3, 0, 1, and 4. she would climb 8 flights.

3 0
2 years ago
(1-5z)(2-5z)=(1−5z)(2−5z)=
miss Akunina [59]

Answer: 0

<u>Simplify both sides of the equation</u>

<u></u>25z^2-15z+2=25z^2-15z+2<u></u>

<u></u>

<u>Subtract 25z² from both sides</u>

<u></u>25z^2-15z+2-25z^2=25z^2-15z+2-25z^2<u></u>

<u></u>-15z+2=-15z+2<u></u>

<u></u>

<u>Add 15z to both sides</u>

<u></u>-15z+2+15z=-15z+2+15z<u></u>

<u></u>2=2<u></u>

<u></u>

<u>Subtract 2 from both sides</u>

<u></u>2-2=2-2<u></u>

<u></u>

Final Answer: 0=0

<u></u>

3 0
3 years ago
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