Base BC of an equilateral triangle ABC is root2 x+y+2=0. If vertex A is the origin, then the area of triangle ABC is (in sq. units) A) 3 root3 B) 2/3 root3 C) 4/3 root3 D) 6 root3
Answers
Answer: Choice C
4/(3*sqrt(3))
which is the same as writing [tex]\frac{4}{3\sqrt{3}}[/tex]
==============================================================
Explanation:
We have the line [tex]\left(\sqrt{2}\right)x+y +2 = 0[/tex] in which points B and C are on this line. In other words, all of segment BC is on that line.
Point A is somewhere off that line as the diagram below indicates. We need to find the line through A that is perpendicular to the given line.
Recall that for any line of the standard form [tex]Px+Qy+R = 0\\\\[/tex], any perpendicular line will look like this: [tex]Qx-Py+S = 0\\\\[/tex]. Note the swap and sign change for the P,Q coefficients.
So anything perpendicular to [tex]\left(\sqrt{2}\right)x+y +2 = 0[/tex] will be of the form [tex]x-\left(\sqrt{2}\right)y+S = 0[/tex]. The sub-goal here is to find what S is equal to.
If you plugged in the origin point (x,y) = (0,0), then you should find that S = 0.
Therefore, the equation [tex]x-\left(\sqrt{2}\right)y = 0[/tex] is perpendicular to the given line and it passes through point A(0,0).
---------------------------
The reason why we computed that perpendicular line is to help determine the point D as shown in the diagram below. This point is at the intersection of the green perpendicular lines. This is at the base of the triangle such that BC is the base.
To find the location of point D, you need to solve this system of equations
[tex]\begin{cases}\left(\sqrt{2}\right)x+y +2 = 0\\\\x-\left(\sqrt{2}\right)y = 0\end{cases}[/tex]
Let's isolate x in the second equation
[tex]x-\left(\sqrt{2}\right)y = 0\\\\x=\left(\sqrt{2}\right)y\\\\[/tex]
Then substitute that into the other equation. Solve for y.
[tex]\left(\sqrt{2}\right)x+y +2 = 0\\\\\left(\sqrt{2}\right)*\left(\sqrt{2}\right)y+y +2 = 0\\\\2y+y +2 = 0\\\\3y +2 = 0\\\\3y = -2\\\\y = -2/3\\\\[/tex]
Now we can find x like so
[tex]x=\left(\sqrt{2}\right)y\\\\x=\left(\sqrt{2}\right)*(-2/3)\\\\x=\frac{-2\sqrt{2}}{3}\\\\[/tex]
The point D is located at the exact coordinates of
[tex]D=(x,y) = \left(\frac{-2\sqrt{2}}{3},\frac{-2}{3}\right)\\\\[/tex]
---------------------------
Next, we use the distance formula to compute the distance between point A and point D.
I'll use lowercase d to represent distance.
[tex]A = (x_1,y_1) = (0,0)\\\\D = (x_2,y_2) = \left(\frac{-2\sqrt{2}}{3},\frac{-2}{3}\right)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{\left(0-\left(\frac{-2\sqrt{2}}{3}\right)\right)^2 + \left(0-\left(\frac{-2}{3}\right)\right)^2}\\\\d = \sqrt{\left(\frac{2\sqrt{2}}{3}\right)^2 + \left(\frac{2}{3}\right)^2}\\\\d = \sqrt{\frac{8}{9} + \frac{4}{9}}\\\\d = \sqrt{\frac{12}{9}}\\\\d = \sqrt{\frac{4}{3}}\\\\d = \frac{\sqrt{4}}{\sqrt{3}}\\\\d = \frac{2}{\sqrt{3}}\\\\[/tex]
This represents the exact length of segment AD.
---------------------------
Since triangle ABC is equilateral, this means triangles ADB and ADC are identical mirrored copies of each other. Those two smaller triangles are equal in area and combine to form triangle ABC back again. Furthermore, they are right triangles. This last fact mentioned will allow us to use the pythagorean theorem like so
[tex]a^2+b^2 = c^2\\\\\left(\text{AD}\right)^2+\left(\text{DC}\right)^2=\left(\text{AC}\right)^2\\\\\left(\frac{2}{\sqrt{3}}\right)^2+\left(\frac{x}{2}\right)^2=\left(x\right)^2\\\\[/tex]
where x represents the lengths of AB, BC and AC.
After skipping a bit of steps, solving that said equation should lead to these two solutions
[tex]x = -\frac{4}{3}, \ x = \frac{4}{3}[/tex]
Because x is a side length, we ignore the negative solution.
---------------------------
We can now compute the area of triangle ABC
[tex]\text{area of triangle} = \frac{1}{2}*\text{base}*\text{height}\\\\\text{area of triangle ABC} = \frac{1}{2}*\text{BC}*\text{AD}\\\\\text{area of triangle ABC} = \frac{1}{2}*\frac{4}{3}*\frac{2}{\sqrt{3}}\\\\\text{area of triangle ABC} = \frac{1*4*2}{2*3*\sqrt{3}}\\\\\text{area of triangle ABC} = \frac{8}{6\sqrt{3}}\\\\\text{area of triangle ABC} = \frac{4}{3\sqrt{3}}\\\\[/tex]
which points us to choice C as the final answer.
When writing this on a keyboard, we would write "4/(3*sqrt(3))" without quotes. The use of parenthesis is needed to ensure we divide by all of 3*sqrt(3). Otherwise, the computer might mistake it for [tex]\frac{4}{3}\sqrt{3}[/tex] which is entirely different.
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Which is steeper & why?
A ramp with a rise of 2ft and a run of 6ft
OR
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Consider the graph of rational function
fo)
6
-6
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2
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O
-2
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A f(3) =
$+ 12
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oc f(3) = =+2
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Answers
Answer:That would be B
Step-by-step explanation:
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options:
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−30
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Answers
Answer:
-30
Step-by-step explanation:
Used a calculator lol
I recently had work done on my car. The estimate was $234.78 but the actual was $365.99.
What is the relative error in the estimate?
Round to 2 decimal places.
Answers
Answer:
35.85%
Step-by-step explanation:
first, find the difference between the real value and the estimated value:
365.99 - 234.78 = 131.21
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How many tenths are there in ⅘
Answers
Answer:
8
Step-by-step explanation:
4/5 = ?/10
4/5 * (2/2) = ?/10
8/10 = ?/10
Answer:
8
Step-by-step explanation:
multiply the denominator and numerator, you get 8/10
is (6,-7) a solution to y+3=1/2(x+2)
Answers
Answer: (4,-2)
To get x substitute 0 for y since you’re trying to find x, 0+3=1/2(x+2) add 0 with 3 to get 3 so 3=1/2(x+2) then use distributive property with 1/2 (x•1/2 = x/2) (2•1/2 = 1) so then the equation would look like this 3=x/2+1 subtract one from both side and get 2=x/2 then multiply everything by 2 to get x=4
With y you can substitute 4 or 0 for x to find your answer
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Answers
Answer:
25-22. 3
---------- = ---
2-1. 1
slope is equal to 3
At the art store, Aisha bought a few blocks of clay to make a teapot and matching mugs for her aunt. She used 2 1/2 pounds of clay to make the mugs and 1 1/2 times as much clay for the teapot. If each block of clay weighed 2 pounds, how many blocks did Aisha use in all?
Answers
13 3/4 is the correct answer
0.3 divided by 6 =
Please help me show my work
Answers
Answer:
0.05
Step-by-step explanation: (this is suppose to be long Division and thats a line between 6 and 0).
0. 0 5
----------------------
6 | 0. 3 0
− 0
----------
0 3
− 0
--------------------------------
3 0
− 3 0
----------------------------------
0
solve pls brainliest!
Answers
Answer:
m - 19 = 58
Step-by-step explanation:
Knowing that the word "difference" relates to subtraction, you should know that the equation involves a minus sign.
The problem also tells us that 58 is the difference. Basically, the word difference means the answer to a subtraction problem. So 58 is the answer to this equation.
However, we are not done yet. It says to use "m" to represent Matt's age. Thus, we can write our equation as:
m - 19 = 58.
The variable "m" goes first instead of 19 because the answer would no longer be 58. For instance, let's say m is 18, 19-18 is equal to 1...not 58.
Therefore, the correct equation is:
m - 19 = 58
Hope this helps!! :D
Who can answer this?
Answers
Answer:
B
Step-by-step explanation:
Explanation: it’s mirroring the other side
Zayn puts garland around a square window. He uses 888 meters of garland.
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Answers
Answer: 3552
Step-by-step explanation:
888 times 4
Find the Measure of The missing angles
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Answer:
∆K = 98°
∆M = 82°
∆G = 118°
∆H = 62°
Step-by-step explanation:
∆K & ∆G is by corresponding angle
and
∆M & ∆H is by linear pairs
Why does that answers show sometimes but not all the time
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Answer:
Don’t be lame
Step-by-step explanation:
Cheaters are lame
Answer:
thats because of brainly ads, if u don't want the brainly ads u have to get premium.
Step-by-step explanation:
Devan stand s926 meters
Answers
Answer:
I believe your missing part of the question.
What is the tenth term of the given geometric sequence: -1, 4, - 16, 64, ... ?
А: 262,144
B: 4,194,304
C: -65,536
D: -1,048,576
Answers
Answer:
262144
Step-by-step explanation:
Geometric sequence : -1 , 4 , -16 , 64 .....
Ratio = 2nd term ÷ 1st term = 4 ÷ (-1)= -4
a = first term = -1
r = -4
Each term is obtained by multiplying the previous term by (-4)
nth term = [tex]ar^{n-1}[/tex]
10th term = [tex](-1) * (-4)^{9}= -1 * (-262144) = 262144[/tex]
A baseball pitcher has pitched 82 2/3 Innings. What is the number of innings written as a decimal?
Answers
82.6666666666666666
The 6’s will go on forever so, in your answer you need to put a horizontal line on top of the six:
Ex. _
82.66
Any of these answers are correct, you can choose which to use. The safest answer would be the last one that has the line on top.
Pls mark brainliest :)
are the two triangles congruent 6cm 13cm
Answers
Congruent triangles are triangles that have equal corresponding angles and sides.
There is no enough information to determine if the triangles are congruent.
From the images of the triangles (see attachment), we can see that the two triangles have 6cm and 13cm as the lengths of their sides.
However, the side lengths of both triangles are not corresponding, and the angle measures are not given.
Hence, there is no enough information to determine if the triangles are congruent.
Read more about congruent triangles at:
https://brainly.com/question/1675117
Solve the inequality: x+10>2x/5. *
Answers
Hello BATAPAAKO!
[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]
Solve the inequality: x+10>2x/5.
[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]
[tex]x + 10 \gt \frac { 2 x } { 5 } \\ [/tex]
Firstly, multiply both sides of the equation by 5.
[tex]5(x + 10) > 2x \\ = 5x + 50 > 2x[/tex]
Since 5 is greater than 0, the direction of the inequality will remain unchanged. Now, subtract 2x from both the sides.
[tex]5x + 50 > 20 \\ = 5x + 50 - 2x > 0[/tex]
Combine 5x & -2x to get 3x.
[tex]5x + 50 - 2x > 0 \\ = 5x - 2x + 50 > 0 \\ = 3x + 50 > 0[/tex]
Now, subtract 50 from both the sides. Remember :- anything subtracted by 0 will result in its negation. So 50 subtracted from 0 will give you -50.
[tex]3x + 50 > 0 \\ = 3x > - 50[/tex]
Divide both sides of the equation by 3. Again, since 3 is > 0, the direction of inequality will remain unchanged.
[tex]3x > - 50 \\ = \large\boxed{\boxed{ \bf \: x > - \frac{ 50}{3} }}[/tex]
__________________
Hope it'll help you!
ℓucαzz ッ
Evaluate a1 (r)^n-1 for a1 = 2, r= 3, and n = 4.
A. 216
B. 18
C. 32
D. 54
Answers
Answer:
[tex]54[/tex]
Step-by-step explanation:
[tex] a_1 {r}^{n - 1} \\ 2 \times {3}^{4 - 1} \\ 2 \times {3}^{3} \\ 2 \times 3 \times 3 \times 3 \\ 2 \times 9 \times 3 \\ 18 \times 3 \\ = 54[/tex]
Hope this helps you.
Let me know if you have any other questions :-):-)
A library classifies at DVDS into one of the cater forums shown. The library owns 1555 DVDS. Determine the number of comedy. 31% comedy
Answers
Answer:
482.05
Step-by-step explanation:
31/100 x 1555 = 482.05
This should be the answer, good luck!
How to simplify the following expressions?
1. 6x-3y+8x-(-4y-3y)
2. 1+8a/4-3+2a
Answers
Answer with Step-by-step explanation:
1. 6x-3y+8x-(-4y-3y)
14x-3y+4y+3y
14x+4y
1+8a/4-3+2a
-2+2a-2a
-2a
Penny reads 10 pages in 1/3 hour. What is the unit rate for pages per hour? For hours per page?
Answers
it would probably be 7, since 1 hour and 30 minutes would be 90 minutes, and she read 10 pages in that long, it would most likely be one page every 10 minutes, but since she read 10 in 90 minutes and not 9, I believe we would add 1 extra page to the amount she read in one hour, and since one hour is 60 minutes, it would probably be 6 + 1, which is 7.
hopefully Im right and this was helpful.
Shawna lives in an apartment 9 4/5 miles from the hospital where she works. Her brother rents a room in a house 7 2/5 miles from the law firm where he is employed. How much farther from work does Shawna live?
Answers
Answer:
2 2/5 miles
Step-by-step explanation:
implify 32 + 3(6 − 22) ÷ 6.
options:
A. 10
B.11
C.4
D.7
Answers
Hey there!
32 + 3(6 − 22) ÷ 6
= 32 + 3(-16) ÷ 6
= 32 + (-48)/6
= 32 + (-8)
= 32 - 8
= 24
Therefore, your answer SHOULD BE: 24
Good luck on your assignment and enjoy your day!
~Amphitrite1040:)
Answer:
24
Step-by-step explanation:
You have to solve this according to BODMAS theorem,
That means,
B - brackets
O - of
D - division
M - multiply
A - addition
S - subtract
Let's solve
32 +3 (6-22) ÷ 6
32 + 3 × (-16) ÷ 6
32 - 48 ÷ 6
32 - 8
24
please help me with this sum
Answers
Answer:
2880
1950
3960
2240
920
5220
4320
3760
6650
1080
32980 is the sum of all combined
Please allow me to know if this helped you by clicking thank you, rating or brainly!
Miss Hawaii
Answer:
1. 2880
2. 1950
3. 3960
4. 2240
5. 920
6. 5220
7. 5760
8. 3760
9. 6650
10. 1080
Need help ASAP. Due in 15min….
Answers
[tex]1 \times \frac{3}{5} + ( \frac{7}{10} )[/tex]
Answer:
1 \times \frac{3}{5} + ( \frac{7}{10} )
Answer:
Second and Fifth answers
Step-by-step explanation:
The negative is being multiplied by a negative so the 7/10 should be positive.
At a bargain store beast boy bought 5 items that each cost the same amount. Cyborg bought 8 items that each cost the same amount, but each was $1.50 less than the items beast boy bought. Both beast boy and Cyborg paid the same amount of money. What was the individual cost of each person's items?
A: Write an equation. Let x represent the cost of one of beast boys items.
B: Solve the equation. Show your work
C: Check your solution. Show your work
D: What was the individual cost of each person's items? State the solution in complete sentences.
Answers
Answer:Bb 12. Cyborg 12
Step-by-step explanation:
Right for my test
The required number of items purchased by the beast boy is 4 and by Cyborg is 2.5.
Given that,
At a bargain store beast boy bought 5 items that each cost the same amount. Cyborg bought 8 items that each cost the same amount, but each was $1.50 less than the items beast boy bought. Both beast boy and Cyborg paid the same amount of money.
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Let the cost of the item bought by the beast boy is x, and the item bought by the Cyborg be y,
According to the question,
5x = 8y
x = 8/5y - - - - - - -(1)
y = x - 1.50 - - - - - -(2)
Substitute the value of equation 1 to equation 2
y = 8/5y - 1.50
5y = 8y - 7.50
7.50 = 3y
y = 2.5
Now, put y in equation 1
x = 8 / 5 × 2.5
x = 8 × 0.5
x = 4
Now, the costs,
5x = 4 × 5 = $20
8y = 8 × 2.5 = $20
Thus, the required number of items purchased by the beast boy is 4 and by Cyborg is 2.5.
Learn more about simplification here: https://brainly.com/question/12501526
#SPJ2
How to tell if a graph is a function in algebra
Answers
Answer:
Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
Step-by-step explanation:
If there is any such line, the graph does not represent a function. If no vertical line can intersect the curve more than once, the graph does represent a function.
