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21 day ago

Triangles ΔABC ≅ ΔBAD so that C and D lie in the opposite semi-planes of segment AB. Prove that segment CD bisects segment AD.

Mathematics

1 answer:

Zina [9.1K]21 day ago

6 0

Answer:

Review the explanation

Step-by-step explanation:

The triangles ΔABC and ΔBAD are congruent, establishing:

AB ≅ BA;
AC ≅ BD;
BC ≅ AD;
∠ABC ≅ ∠BAD;
∠BCA ≅ ∠ADB;
∠CAB ≅ ∠DBA.

Now, consider triangles AEC and BED. In these triangles, we have:

AC ≅ BD;
∠EAC ≅ ∠EBD (due to ∠CBA ≅ ∠BAD);
∠AEC ≅ ∠BED (being vertical angles).

Therefore, ΔAEC ≅ ΔBED, which leads to the conclusion that:

AE ≅ EB.

This indicates that line segment CD bisects segment AD.

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Answer:

a) 20 different arrangements

b) 10 different arrangements

Step-by-step explanation:

When the order of selection is significant, we utilize permutations to determine the number of arrangements; if the order does not matter, we apply combinations.

For part a)

We must select 2 items from a collection of 5, considering that order matters. This is a permutation scenario, thus we calculate 5P2.

The general formula for permutations of n items taken r at a time is:

$nPr=\frac{n!}{(n-r)!}$

By inserting n=5 and r=2, we have:

$5P2=\frac{5!}{(5-2)!} =20$

Hence, it is possible to select 2 items from a group of 5 when the selection order is relevant.

For part b)

Since the order of selection is unimportant, we use combinations to calculate the number of ways of selecting 2 items from 5, represented as 5C2.

The general formula for combinations of n items taken r at a time is:

$nCr=\frac{n!}{r!(n-r)!}$

Plugging n=5 and r=2 into the formula results in:

$5C2=\frac{5!}{2!(5-2)!} =10$

This demonstrates that we can choose 2 items from a collection of 5 when the order of selection is not significant.

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a desert has both fruit and yogurt inside altogether the mass of the desert is 185g the ratio of the mass to fruit to the mass o

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Fruit: yoghurt: total
2: 3: 2 + 3
2: 3: 5

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The weight of the yoghurt is 111g

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Answer:

a) Slope: $\hat \beta_1 =\frac{7625.9}{1248.9}=6.106$

Intercept: $\hat \beta_o = 157.955 -6.106 (69.686)=-267.548$

b) $r=\frac{7625.9}{\sqrt{[1248.9][94228.8]}}=0.657$

Additionally, the coefficient of determination is $r^2 = 0.657^2 =0.432$

Step-by-step explanation:

Definitions and data provided

The correlation coefficient is a measure that quantifies the strength of the relationship between two variable movements, denoted as r and ranging between -1 and 1.

The sum of squares refers to the total of the squared deviations, where deviation is defined as the difference between each value and the grand mean.

When performing multiple regression, the aim is to analyze the relationship between multiple independent variables and one dependent variable.

n=110, $\sum x_i y_i = \sum (X-\bar X)(Y-\bar Y) =7625.9,\sum x^2_i=\sum (x-\bar x)^2 =1248.9, sum y^2_i=\sum(y-\bar y)^2 =94228.8$

$\sum Y_i =17375, \sum X_i = 7665.5$

Part a

The slope can be calculated using this formula:

$\hat \beta_1 =\frac{\sum (x-\bar x) (y-\bar y)}{\sum (x-\bar x )^2}$

Following the substitutions, we have:

$\hat \beta_1 =\frac{7625.9}{1248.9}=6.106$

The intercept can be determined with this formula:

$\hat \beta_o = \bar y -\hat \beta_1 \bar x$

Average values for x and y can be calculated this way:

$\bar X=7665.5/110 =69.686, \bar y= 17375/110=157.955$

Replacing yields:

$\hat \beta_o = 157.955 -6.106 (69.686)=-267.548$

Part b

The correlation coefficient can be calculated using the following formula:

$r=\frac{\sum (x-\bar x)(y-\bar y) }{\sqrt{[\sum (x-\bar x)^2][\sum(y-\bar y)^2]}}$

In our situation:

n=110, $\sum x_i y_i = \sum (X-\bar X)(Y-\bar Y) =7625.9,\sum x^2_i=\sum (x-\bar x)^2 =1248.9, sum y^2_i=\sum(y-\bar y)^2 =94228.8$

We can compute the correlation coefficient by substituting values:

$r=\frac{7625.9}{\sqrt{[1248.9][94228.8]}}=0.657$

The coefficient of determination is $r^2 = 0.657^2 =0.432$

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