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VladimirAG [237]
3 years ago
14

Prove that a parallelogram is a square iff its diagonals are both congruent and perpendicular

Mathematics
1 answer:
Juliette [100K]3 years ago
6 0
Consider the parallelogram shown below.
The lengths of the sides are a and b.
The lengths of the diagonals are 2x and 2y.

Because the diagonals are both congruent and perpendicular, therefore there are two right triangles as shown.
Note that x = y.
Because x = y, each right triangle is isosceles and has the angles 90°, 45° and 45°.

From the Pythagorean theorem,
For one right triangle,
a² = x² + y² = x² + x² = 2x².
For the other right triangle,
b² = y² + x² = x² + x² = 2x²

Therefore
a² = b²
a = b

It follows that all sides of the parallelogram are equal and each angle is
45+45 = 90°

Therefore the parallelogram is a square.

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Use the figure to decide the type of angle pair that describes ∠3 and ∠6.
allochka39001 [22]

Answer:

Same Side Exterior Angles

Step-by-step explanation:

They are on the same side of the line and both angles are exterior of the other two lines.

4 0
3 years ago
Which of the following exponential regression equation best fits the data shown below. Please help ASAP. ☺️
stepladder [879]

Answer:

Option D [y=6.61\,*\,1.55^x] in the list of possible answers

Step-by-step explanation:

For this problem you are supposed to use a calculator that allows you to do an exponential regression. There are many tools that can help you with that, depending on what your instructors has assigned for your class.

I am showing you the results of a graphing tool I use, and which after entering the x-values and the y-values in independent "List" forms, when I request the exponential regression to fit the data, I get what you can see in the attached image.

Notice that the exponential of best fit with my calculator comes in the form:

y=A\,e^{k\, x}

with optimized parameters:

A \approx 6.6114\,\,\,and\,\,\, k=0.4378321

Notice as well that since:

e^{0.4378321} \approx 1.5490

the exponential best fit can also be written:

y=6.611403\,\,*\,e^{0.4378321\, x}=6.611403\,*\,1.549^{\,x}

and this expression is very close to the last option shown in your list of possible answers

8 0
3 years ago
Nevermind about the other one this is the one that’s due today: Julio wears a blue shirt every 4 days. Larry wears a blue shirt
cupoosta [38]

Answer:

Larry will wear one on April 15 and Julio will wear one on april 14

Step-by-step explanation:

5 0
2 years ago
Read 2 more answers
$15.30 divided by 15??????
alekssr [168]
The answer is 1.02 according to calculations
8 0
3 years ago
Read 2 more answers
Construct a​ 99% confidence interval for the population​ mean, mu. Assume the population has a normal distribution. A group of 1
Zarrin [17]

Answer:

99% confidence interval for the population​ mean is [19.891 , 24.909].

Step-by-step explanation:

We are given that a group of 19 randomly selected students has a mean age of 22.4 years with a standard deviation of 3.8 years.

Assuming the population has a normal distribution.

Firstly, the pivotal quantity for 99% confidence interval for the population​ mean is given by;

         P.Q. = \frac{\bar X - \mu}{\frac{s}{\sqrt{n} } } ~ t_n_-_1

where, \bar X = sample mean age of selected students = 22.4 years

             s = sample standard deviation = 3.8 years

             n = sample of students = 19

             \mu = population mean

<em>Here for constructing 99% confidence interval we have used t statistics because we don't know about population standard deviation.</em>

So, 99% confidence interval for the population​ mean, \mu is ;

P(-2.878 < t_1_8 < 2.878) = 0.99  {As the critical value of t at 18 degree of

                                                freedom are -2.878 & 2.878 with P = 0.5%}

P(-2.878 < \frac{\bar X - \mu}{\frac{s}{\sqrt{n} } } < 2.878) = 0.99

P( -2.878 \times {\frac{s}{\sqrt{n} } } < {\bar X - \mu} < 2.878 \times {\frac{s}{\sqrt{n} } } ) = 0.99

P( \bar X -2.878 \times {\frac{s}{\sqrt{n} } < \mu < \bar X +2.878 \times {\frac{s}{\sqrt{n} } ) = 0.99

<u>99% confidence interval for</u> \mu = [ \bar X -2.878 \times {\frac{s}{\sqrt{n} } , \bar X +2.878 \times {\frac{s}{\sqrt{n} } ]

                                                 = [ 22.4 -2.878 \times {\frac{3.8}{\sqrt{19} } , 22.4 +2.878 \times {\frac{3.8}{\sqrt{19} } ]

                                                 = [19.891 , 24.909]

Therefore, 99% confidence interval for the population​ mean is [19.891 , 24.909].

6 0
3 years ago
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