All categories

3 years ago

These 3 please!!! Properties of Kites and Trapzoids exercises 1-3

Mathematics

1 answer:

scoundrel [369]3 years ago

7 0

Answer:

area = 1/2(AC)(BD)
yes; BE can be found using the Pythagorean theorem
yes

Step-by-step explanation:

Triangles ADB and CDB are congruent, so each is half the total area. The area of ADB is ...

A = 1/2bh

A = 1/2(DB)(1/2AC) = 1/4(DB)(AC)

The area of the kite is twice this value:

A = 1/2(DB)(AC)

In order to use the formula derived in part 1, we need to know the length of BE. That can be found using the Pythagorean theorem:

BE² +AE² = AB²

We know that AE = 1/2AC, so the desired dimension is ...

BE = √(AB² -(1/4)AC²)

Then the area of the kite is ...

area = 1/2(DB)(AC)

area = (1/2)(AC)(ED +√(AB² -(1/4)AC²)

Given three sides of a triangle, the area of it can be found using Heron's formula. If you have not yet seen that in your geometry class, you can always use the Pythagorean theorem and some algebra.

As before, we need to know the lengths of the diagonals. One is given. The other can be found considering the Pythagorean relations:

AD² = AE² +DE²

AB² = AE² +BE²

BE +DE = BD . . . . the sum of the horizontal diagonal lengths

Subtracting the second equation from the first gives ...

AD² -AB² = DE² -BE²

Using the third to write an expression for DE, we get an equation for BE:

AD² -AB² = (BD -BE)² -BE²

AD² -AB² = BD² -2BD·BE +BE² -BE²

(AB² +BD² -AD²)/(2BD) = BE

Substituting this into the original second equation, we can find AE, hence the area of the kite.

AE² = AB² -((AB² +BD² -AD²)/(2BD))²

Which means the kite area is ...

area = BD·AE = BD√(AB² -((AB² +BD² -AD²)/(2BD))²)

area = (1/2)√(4·AB²·BD² -(AB² +BD² -AD²)²

This area formula demonstrates it is possible to find the area of the kite given the three lengths AB, BD, DA.

_____

<em>Additional comment</em>

We have no doubt that the area formula of Part 3 can be rearranged to match Heron's formula. Here, we're interested in demonstrating area can be found.

You might be interested in

3•3•3 in exponential form

shtirl [24]

3 with a little 3 in the top right corner because your are multiplying 3 times

5 0

4 years ago

Read 2 more answers

From 19 measurements of a pressure controller, the mean operating pressure is 4.97 kPa with a standard deviation of 0.0461 kPa.

Semenov [28]

Answer:

$4.97-2.10\frac{0.0461}{\sqrt{19}}=4.95$

$4.97+2.10\frac{0.0461}{\sqrt{19}}=4.99$

So on this case the 95% confidence interval would be given by (4.95;4.99)

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X =4.97$ represent the sample mean

$\mu$ population mean (variable of interest)

s=0.0461 represent the sample standard deviation

n=19 represent the sample size

Confidence interval

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=19-1=18$

Since the Confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,18)".And we see that $t_{\alpha/2}=2.10$