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3 years ago

Write a two column proof. If ZY=3XY, then ZX=4XY

Mathematics

1 answer:

Blababa [14]3 years ago

5 0

Answer:

As you have written

Z Y = 3 X Y,

It means X, Y, Z are collinear points i.e they lie in the same line or line segment.

Locating points X, Y,Z on the line

Two possibilities are possible

1. First Z, then X and then Y on the line or line segment

→ Z X + Y X= Z Y

But Z Y = 3 X Y [ Given]

→ Z X + Y X = 3 X Y

→ Z X = 3 X Y - Y X

→ Z X = 2 X Y ⇒ which is not the result.

2. 2nd Possibility

First Z, then Y and then X on the line or line segment

→ Z X = Z Y + Y X

→ Z X= 3 X Y + Y X [Z Y=3 X Y→(given)]

→ Z X= 4 X Y , Which is the result.

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= 198

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3 years ago

78, 95, 77, 92, 87, 79, 94, 86, 88, 94 <br>Mean:<br>Median:<br>Range:<br>Mean absolute deviation:

Oxana [17]

<h2>Explanation:</h2>

The mean is a measure of central tendency, also called the <em>average. </em>It is found as:

$A ={\frac{1}{n} \sum_{i=1}^n (x_i)$

$Where: \\ \\ n:\text{Total number of values} \\ \\ x_{1}:\text{the individual numbers in the data set.}$

Then:

$A=\frac{78+95+77+92+87+79+94+86+88+94}{10}=\frac{870}{10} \\ \\ \boxed{A=87}$

<h3>Median:</h3><h3 />

The median is a measure of central tendency. It's the value of the center when the data set is sorted.

If the number of observations is odd the formula is given by:

$x_m = x_\frac{n+1}{2}$

If the number of observations is even the formula is given by:

$x_m = \frac{x_\frac{n}{2}+x_\frac{n+2}{2}}{2}$

Since the number of observations is even, we'll use the second formula. The sorted data set is:

$77,78,79,86,87,88,92,94,94,95$

Therefore:

$x_m = \frac{x_\frac{n}{2}+x_\frac{n+2}{2}}{2} \\ \\$

$x_\frac{n}{2}}=x_{5}=87 \\ \\ x_\frac{n+2}{2}}=x_{6}=88 \\ \\ \\ x_m =\frac{87+88}{2} \\ \\ x_{m}=\frac{175}{2} \\ \\ \boxed{x_{m}=87.5}$

<h3>Range:</h3><h3 />

The Range is a measure of dispersion. It tells us by <em>how much the values in the data set are likely to differ from their mean. </em>It's found by subtracting the lowest from the highest value in the data set:

$R=x_{M}-x_{m} \\ \\ \\ Where: \\ \\ R:Range \\ \\ x_{M}:Highest \ value \\ \\ x_{m}:Lowest \ value$

From the data set:

$x_{m}= 77 \\ \\ x_{M}=95 \\ \\ \\ So: \\ \\ R=95-75 \\ \\ \boxed{R=18}$

<h3>Mean absolute deviation:</h3><h3 />

The mean deviation is a <em>measure of dispersion. </em>The formula is given by:

$MAD =\frac{1}{n} \sum_{i=1}^n |x_i-\bar{x}|$

So:

$\bar{x}:Mean \\ \\ x_{i}:each \ value \ in \ the \ data \ set$

The sorted data set:

$77,78,79,86,87,88,92,94,94,95$

So;

$|x_1-\bar{x}|= |77-87|=|-10|=10 \\ \\ |x_2-\bar{x}|= |78-87|=|-9|=9 \\ \\ |x_3-\bar{x}|= |79-87|=|-8|=8 \\ \\ |x_4-\bar{x}|= |86-87|=|-1|=1 \\ \\ |x_5-\bar{x}|= |87-87|=|0|=0 \\ \\ |x_6-\bar{x}|= |88-87|=|-10|=1 \\ \\ |x_7-\bar{x}|= |92-87|=5|=5 \\ \\ |x_8-\bar{x}|= |94-87|=|7|=7 \\ \\ |x_9-\bar{x}|= |94-87|=|7|=7 \\ \\ |x_10-\bar{x}|= |95-87|=|8|=8 \\ \\$

So:

$MAD=\frac{1}{10}(10+9+8+1+0+1+5+7+7+8) \\ \\ \boxed{MAD=5.6}$

5 0

2 years ago