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3 years ago

In trapezoid ABCD, AC is a diagonal and ∠ABC≅∠ACD. Find AC if the lengths of the bases BC and AD are 12m and 27m respectively.

Mathematics

1 answer:

xz_007 [3.2K]3 years ago

4 0

Answer:

Length of diagonal is 18 m

Step-by-step explanation:

Given in trapezoid ABCD. AC is a diagonal and ∠ABC≅∠ACD. The lengths of the bases BC and AD are 12m and 27m. We have to find the length of AC.

Let the length of diagonal be x m

In ΔABC and ΔACD

∠ABC=∠ACD (∵Given)

∠ACB=∠CAD (∵Alternate angles)

By AA similarity theorem, ΔABC~ΔACD

∴ their corresponding sides are proportional

$\frac{x}{27}=\frac{12}{x}=\frac{AB}{CD}$

Comparing first two, we get

⇒ $\frac{x}{27}=\frac{12}{x}$

⇒ $x^2=27\times 12$

⇒ $x=\sqrt324=18$

hence, the length of diagonal is 18 m

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Can someone please help

mrs_skeptik [129]

For the points in the parabola, we have:

A = (1, 0)
B = (3, 0)
P = (0, 3)
Q = (2, -1).

<h3>How to identify the points on the parabola?</h3>

Here we have the quadratic equation:

y = (x - 1)*(x - 3)

First, we want the coordinates of A and B, which are the two zeros of the parabola.

Because it is already factorized, we know that the zeros are at x = 1 and x = 3, so the coordinates of A and B are:

A = (1, 0)

B = (3, 0).

Then point P is the y-intercept, to get it, we need to evaluate in x = 0.

y = (0 - 1)*(0 - 3) = (-1)*(-3) = 3

Then we have:

P = (0, 3)

Finally, point Q is the vertex. The x-value of the vertex is in the middle between the two zeros, so the vertex is at x = 2.

And the y-value of the vertex is:

y = (2 - 1)*(2 - 3) = 1*(-1) = -1

So we have:

Q = (2, -1).

If you want to learn more about quadratic equations:

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2 years ago

Find the given derivative by finding the first few derivatives and observing the pattern that occurs. . . d^114 /dx^114 of (sinx

uranmaximum [27]

So in your given pattern, you need to find first the derivatives and observe the patter that occurs in the given functions. So with this kind of pattern, every fourth one is the same; that makes the 114th derivative is the same as the second derivative. It is known since 114/4 has a remainder of two

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3 years ago

Read 2 more answers

If the value of third order determinant is 11. Then what is the value of the determinant formed by its cofactors.

nordsb [41]

Answer:

146.41

Step-by-step explanation:

third order determinant = determinant of 3×3 matrix A

given ∣A∣=11

det (cofactor matrix of A) =set (transpare of cofactor amtrix of A) (transpare does not change the det)

=det(adjacent of A)

{det (cofactor matrix of A)} ^2 = {det (adjacent of A)} ^2

(Using for an n×n det (cofactor matrix of A)=det (A)^n−1 )

we get

det (cofactor matrix of A)^2 = {det(A) ^3−1 }^2

=(11)^2×2 = 11^4

=146.41

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3 years ago

A parabola passes through the points (-2,8), (0,2), and (1,5). What function does the graph represent?

zheka24 [161]

ANSWER

$y = 2 {x}^{2} + x + 2$

EXPLANATION

Let the function that represent the graph be:

$y = a {x}^{2} + bx + c$

The parabola passes through the points (-2,8), (0,2), and (1,5).

These points must satisfy the function.

For (-2,8), we have

$8= a{( - 2)}^{2} + b( - 2) + c$

This implies that that,

$4a - 2b + c = 8...(1)$

For (0,2), we have,

$2= a{( 0)}^{2} + b( 0) + c$

This implies that,

$c = 2$

For (1,5), we have

$5= a{( 1)}^{2} + b( 1) + c$

This implies that,

$a + b + c = 5...(2)$

Put c=2 into equation (1) and (2).

$4a - 2b + 2 = 8$

$4a - 2b = 8 - 2$

$4a - 2b = 6$

$2a - b = 3...(3)$

$a + b + 2=5$

$a + b =5 - 2$

$a + b = 3...(4)$

Add equation (3) and equation (4)

$3a = 6$

$a = 2$

Put a=2 into equation (4).

$2 + b = 3$

$b = 3 - 2 = 1$

Therefore the function is

$y = 2 {x}^{2} + x + 2$

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4 years ago

For each of the systems of equations that follow, use Gaussian elimination to obtain an equivalent system whose coefficient matr

noname [10]

Answer:

b) inconsistent

d) inconsistent

f) inconsistent

j) X_1 = 2 - 6a , X_2 = 4 + a , X_3 = 3 - a, X_4 = a (free variable)

Step-by-step explanation:

Given: Augmented matrix

$\left[\begin{array}{ccc}2&-3&5\\-4&6&8\end{array}\right]$

Row operation : R_2 + 2*R_1

$\left[\begin{array}{ccc}2&-3&5\\0&0&18\end{array}\right]$

Looking at R_2 : 0 not equal to 18 ; Hence, inconsistent

Given: Augmented matrix

$\left[\begin{array}{cccc}3&2&-3&4\\1&-2&2&1\\11&2&1&14\end{array}\right]$

Row operation : R_2 <---> R_1

$\left[\begin{array}{cccc}1&-2&2&1\\3&2&-3&4\\11&2&1&14\end{array}\right]\\$

Row operations : R_2 - 3*R_1 & R_3 - 11*R_1

$\left[\begin{array}{cccc}1&-2&2&1\\0&8&-9&1\\0&24&-21&3\end{array}\right]\\$

Row operations : R_3 - 3*R_2

$\left[\begin{array}{cccc}1&-2&2&1\\0&8&-9&1\\0&0&6&0\end{array}\right]\\$

Looking at R_3 : 0 not equal to 6 ; Hence, inconsistent

Given: Augmented matrix

$\left[\begin{array}{cccc}1&-1&2&4\\2&3&-1&1\\7&3&4&7\end{array}\right]\\$

Row operations : R_2 - 2*R_1 & R_3 - 7*R_1

$\left[\begin{array}{cccc}1&-1&2&4\\0&5&-5&-7\\0&10&-10&-21\end{array}\right]\\$

Row operations : R_2 --> ( 1 / 5 )* R_2 & R_3 --> ( 1 / 10 )* R_3

$\left[\begin{array}{cccc}1&-1&2&4\\0&1&-1&-1.4\\0&1&-1&-2.1\end{array}\right]\\$

Row operations : R_3 - R_1

$\left[\begin{array}{cccc}1&-1&2&4\\0&1&-1&-1.4\\0&0&0&-0.7\end{array}\right]\\$

Looking at R_3 : 0 not equal to -0.7 ; Hence, inconsistent

Given: Augmented matrix

$\left[\begin{array}{ccccc}1&2&-3&1&1\\-1&-1&4&-1&6\\-2&-4&7&-1&1\end{array}\right]\\$

Row operations : R_2 + R_1 & R_3 + 2*R_1

$\left[\begin{array}{ccccc}1&2&-3&1&1\\0&1&1&0&7\\0&0&1&1&3\end{array}\right]\\$

corresponding equations are:

X_1 + 2*X_2 - 3*X_3 + X_4 = 1

X_2 + X_3 = 7

X_3 + X_4 = 3

Free parameter : X_4 = a , then:

X_3 = 3 - a

X_2 = 4 + a

X_1 = 1 - a + 3*(3 - a) - 2*(4 + a) = 2 - 6 a

4 0

4 years ago