Points M, N, and P are respectively the midpoints of sides AC , BC , and AB of △ABC. Prove that the area of △MNP is on fourth of the area of △ABC.
Answers
Answer:
The area of △MNP is one fourth of the area of △ABC.
Step-by-step explanation:
It is given that the points M, N, and P are the midpoints of sides AC, BC and AB respectively. It means AC, BC and AB are median of the triangle ABC.
Median divides the area of a triangle in two equal parts.
Since the points M, N, and P are the midpoints of sides AC, BC and AB respectively, therefore MN, NP and MP are midsegments of the triangle.
Midsegments are the line segment which are connecting the midpoints of tro sides and parallel to third side. According to midpoint theorem the length of midsegment is half of length of third side.
Since MN, NP and MP are midsegments of the triangle, therefore the length of these sides are half of AB, AC and BC respectively. In triangle ABC and MNP corresponding side are proportional.
[tex]\triangle ABC \sim \triangle NMP[/tex]
[tex]MP\parallel BC[/tex]
[tex]MP=\frac{BC}{2}[/tex]
By the property of similar triangles,
[tex]\frac{\text{Area of }\triangle MNP}{\text{Area of }\triangle ABC}=\frac{PM^2}{BC^2}[/tex]
[tex]\frac{\text{Area of }\triangle MNP}{\text{Area of }\triangle ABC}=\frac{(\frac{BC}{2})^2}{BC^2}[/tex]
[tex]\frac{\text{Area of }\triangle MNP}{\text{Area of }\triangle ABC}=\frac{1}{4}[/tex]
Hence proved.
The area of triangle MNP is one-fourth that of triangle ABC, because M, N, and P are midpoints leading to the smaller triangle being similar to the larger, and thus their area ratio due to the proportional side lengths is 1:4.
Explanation:In Mathematics, proving that the area of △MNP is a fourth of that of △ABC involves the concept of similar triangles and their properties. We know that M, N, and P are the midpoints of BC, AC, and AB. Due to this, triangle MNP is similar to triangle ABC as per the Midpoint theorem. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. In this case, the ratio of the sides would be 1:2 (since M, N, P are midpoints).
Therefore, the ratio of their areas (△MNP:△ABC) would be (1⁄2)², which is 1/4. Hence, the area of △MNP is one-fourth of the area of △ABC.
This clearly demonstrates that the area of the smaller triangle formed by the midpoints of a larger triangle equates to a fourth of the total area of the initial triangle.
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Related Questions
Nick purchased a box of 425 thumbtacks. The box contained green thumbstacks and yellow thumbstacks. The number of green thumbstacks to the number of yellow thumbstacks was in the ratio of 7:10. How many yellow thumbstacks were in this box?
Answers
Answer:
250 yellow thumbtacks
Step-by-step explanation:
7:10 ratio (17 thumbtacks)
425 / 17 = 25
7 x 25 = 175 green
10 x 25 = 250 yellow
what is 152% of 72 plz give me answer
Answers
Answer:
152% of 72 = 109.44
Step-by-step explanation:
is Y=2 to the 2nd power linear or non linear why?
Answers
Answer:
Linear
Step-by-step explanation:
y = 2^2
y = 4
This is a line. 4 is constant.
Y to a constant value is a horizontal line
Consider the function f(x)=−2/3x+5 . What is f(12) ? Enter your answer, as a simplified fraction, in the box.
Answers
Answer:
Step-by-step explanation:
f(12)=-2/3(12)+5
f(12)=-8+5
f(12)=-3
[tex]f(x)=-\frac{2}{3}x+5[/tex]
f(12) this means that x is 12, so you can plug in 12 for "x" in the function:
[tex]f(x)=-\frac{2}{3}x+5[/tex]
[tex]f(12)=-\frac{2}{3}(12)+5[/tex]
[tex]f(12)=-\frac{24}{3} +5[/tex]
f(12) = -8 + 5
f(12) = -3
Your answer is -3
-2⁄3 a − 6 = 12
a =
Answers
Answer:
a = -27
Step-by-step explanation:
-2⁄3 a − 6 = 12
add 6 to both sides:-
-2/3 a -6 + 6 = 12 + 6
-2/3 a = 18
Now multiply each side by -3/2:-
a = 18 * -3/2 = -54/2
= -27
To solve the equation -2⁄3 a - 6 = 12 for 'a', add 6 to both sides of the equation and then multiply both sides of the equation by -3/2 to isolate the variable 'a'. The value of 'a' that satisfies the equation is a = -27.
Explanation:To solve the equation -2⁄3 a - 6 = 12 for 'a', we can start by isolating the variable on one side of the equation. First, let's add 6 to both sides of the equation:
-2⁄3 a - 6 + 6 = 12 + 6
-2⁄3 a = 18
Next, we can multiply both sides of the equation by -3/2 to get:
(-3/2)(-2/3 a) = (-3/2)(18)
a = -27
The value of 'a' that satisfies the equation -2⁄3 a - 6 = 12 is a = -27.
What is the formula for calculating taxable income?
Answers
Which expression represents the approximate length of bc?
Answers
Hello from MrBillDoesMath!
Answer:
BC = 3 * sin (66)/ sin(38)
Discussion:
From the law of sines:
sin(38) /3 = sin(66)/ BC
BC * ( sin(38)/3 ) = sin(66) => multiply both sides by BC
BC = sin(66) / ( sin(38)/3) => multiply both sides by ( 3/sin(38) )
BC = 3 * sin (66)/ sin(38)
which is the first choice.
Regards,
MrB
P.S. I'll be on vacation from Friday, Dec 22 to Jan 2, 2019. Have a Great New Year!
Answer:
answer is A
Step-by-step explanation:..
HELPP plzzzzz plzzzz
Answers
Answer:
121/100
Hope it helps!
=> we multiply with 100
1,21= 121/100
simple.
Jose travels 130 miles in 150 minutes. How many miles does Jose travel in 1 hour?
Answers
Answer:
52 miles
Step-by-step explanation:
We can use a ratio to solve this problem. Let put miles over time. We know that 1 hour = 60 minutes. We need to have the units the same.
130 miles x miles
-------------- = ---------------
150 minutes 60 minutes
Using cross products
130 * 60 = 150 x
Divide each side by 150
130/150 * 60 = 150x/150
52 =x
Describe and correct the error in rewriting the equation
2x - y=5
Y= -2x+5
Answers
2x - y = 5 subtract 2x from both sides
2x - 2x - y = 5 - 2x
- y = 5 - 2x
The sign "-" (minus) before y was lost.
My teacher never taught us this, but expects us to know
Answers
I'll do problem 77 to get you started
Point slope form is this general equation
y - y1 = m(x - x1)
with m as the slope and (x1,y1) as the point the line goes through. In this case, m = -4 and (x1,y1) = (-2,9)
Plug those values into the equation to get...
y - y1 = m(x - x1)
y - 9 = -4(x - (-2))
y - 9 = -4(x + 2) see note 1
y - 9 = -4x - 8
y = -4x - 8 + 9
y = -4x + 1 see note 2
--------
note 1: If your teacher wants the answer in point slope form, then you stop on this line. Otherwise, keep going
note 2: This is in slope intercept form y = mx+b with m = -4 as the slope and b = 1 as the y intercept
you’re score on a game show decreases by 5 points for each question you answer incorrectly. You answer 4 questions incorrectly, what integer represents the change in your score?
Answers
Answer:
-20
Step-by-step explanation:
If we get -5 points for each question you answer incorrectly, and we answer 4 questions incorrectly, than we can simply multiply -5 by 4 to get -20.
Answer:
(-20) is an integer represents the change in score.
Step-by-step explanation:
Penalty for answering a question incorrectly = -5
Number of questions answered incorrectly = 4
If we had answered 4 question incorrectly then the penalty of marks will be:
[tex]-5\times 4=-20[/tex]
(-20) is an integer represents the change in score.
PLEASE PLEASE PLEASE HELP! WILL BE GIVING OUT BRAINLIEST AND LOTS OF POINTS!!!!!!!
Gina looks at the architectural plan of a four-walled room in which the walls meet each other at right angles. The length of one wall in the plan is 17 inches. The length of the diagonal of the floor of the room in the plan is approximately 18.79 inches. Is the room in the shape of a square? Explain how you determined your answer. Show all your work.
Answers
Final answer:
The room is not a square, as shown by applying the Pythagorean theorem, because the calculated diagonal of a square with 17-inch sides would be approximately 24.04 inches, not 18.79 inches.
Explanation:
To determine if the room is in the shape of a square, we can use the Pythagorean theorem, which relates the lengths of the sides of a right-angled triangle. For a square, the lengths of all four sides are equal. Therefore, if we assume that the length and the width of the room are both equal to the given wall length of 17 inches, the diagonal (d) of the square can be calculated using the Pythagorean theorem:
c2 = a2 + b2
where c is the length of the diagonal, and a and b are the lengths of the sides of the square.
So, it would be:
c2 = 172 + 172
= 289 + 289
= 578
Now we find the square root of 578 to find the length of the diagonal (c):
c ≈ √578 ≈ 24.04 inches
Since the given diagonal length is approximately 18.79 inches, and our calculated value for the diagonal of a square with sides 17 inches is approximately 24.04 inches, the room is not a square. The diagonal is too short for the room to be a square given the 17-inch wall.
What is the reference angle in degrees for 31π 6 ?
Answers
The reference angle for [tex]31\pi[/tex] 6 by conversion of radians to degrees is [tex]210^o[/tex].
One radian is defined as the angle formed by the center of a circle intersecting an arc of length equal to the radius of the circle.
There are [tex]\pi[/tex] radians in [tex]180^o[/tex], So,[tex]\dfrac{31\pi}{6}[/tex] must be multiplied by [tex]\dfrac{180}{\pi}[/tex]to obtain the angle in degrees.
[tex]\dfrac{31\p}{6} \times \dfrac{180}{\pi}[/tex]
= [tex]\dfrac{31 \times 180}{6}[/tex]
=[tex]31 \times 30[/tex]
= [tex]930^o[/tex]
Since the reference angle is always positive and is formed with the x-axis, we need to find the acute angle formed between the terminal side of the angle and the positive x-axis.
The reference angle can be calculated by subtracting the closest multiple [tex]180^o[/tex] from the angle in degrees.
[tex]930^o - 2\times 180^o[/tex]
= [tex]930^o - 360^o[/tex]
=[tex]570^o[/tex]
However, the reference angle should always be between [tex]0^o[/tex] and [tex]180^o[/tex] So, we take the smaller angle, which is
[tex]570^o- 360^o[/tex]
= [tex]210^o[/tex]
Therefore, the reference angle in degrees [tex]31\pi6[/tex] is [tex]210^o[/tex]
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The reference angle of the given angle is [tex]30\textdegree.[/tex]
Converting the given angle from radians to degrees:
We know that [tex]\pi[/tex] radians is equal to 180°. Therefore,
[tex]\frac{31\pi}{6}\ radians \times \frac{180\textdegree}{\pi} = \frac{31}{6}\times 180 = 930\textdegree[/tex]
Finding the equivalent angle between 0° and 360°:
Since 930° is greater than 360°, an equivalent angle needs to be found by subtracting multiples of 360° until an angle between 0° and 360° is obtained.
930°−2×360°=930°−720°=210°
Determining the reference angle:
The reference angle is the acute angle that the given angle makes with the x-axis. Since 210° is in the third quadrant, calculating the reference angle as:
210°−180°=30°
Therefore, the reference angle for [tex]\frac{31\pi}{6}[/tex] is 30°.
how to divide numbers by decimals
Answers
Multiply the denominators. Way 2: Move decimal points, divide, and place a decimal point in the quotient. Step 1 Write a decimal point and a zero in the dividend. Step 2 Multiply the divisor by a power of 10 to get a whole number.
which property is used to simplify the expression 1+(6+0)=1+6
additive inverse property
additive identity property
associative property of addition
commutative property of addition
Answers
Answer:
commutative property of addition
Step-by-step explanation:
commutative property of addition means that both sides on the equal sign have the same numbers but just has is in a deferent order. It might be confusing for this one because there is a 0, but 0 is nothing so it still applies.
Hope this helps, have a great day!
Solve for x.
.
Simplify your answer as much as possible.
Answers
2x = 15 - 3x -- add 3x to both sides
5x = 15 -- divide both sides by 5
x = 3 -- simplest form
For this case, we must find the value of "x" of the following equation:
[tex]2x = 15-3x[/tex]
We add 3x to both sides of the equation:
[tex]2x + 3x = 15-3x + 3x\\5x = 15[/tex]
We divide between 5 on both sides of the equation:
[tex]\frac {5x} {5} = \frac {15} {5}\\x = 3[/tex]
Thus, the value of x is 3.
Answer:
[tex]x = 3[/tex]
what is the distance between points (2,-3) and (0,8) to the nearest tenth
Answers
The distance between the points (2,-3) and (0,8) can be calculated using the distance formula which derives from the Pythagorean theorem. After plugging in the coordinates into the formula, we get the distance approximately equal to 11.2.
Explanation:To calculate the distance between the points (2,-3) and (0,8), we can make use of the distance formula derived from the Pythagorean theorem. The distance formula is: d = √[(x2-x1)² + (y2-y1)²].
Here, (x1, y1) = (2, -3) and (x2, y2) = (0, 8).
Now, plug the coordinates into the formula:
d = √[(0 - 2)² + (8 - -3)²]
d = √[(-2)² + (11)²]
d = √[4 + 121]
d = √125
So, the distance to the nearest tenth between (2,-3) and (0,8) is approximately 11.2.
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What type of transformation takes the graph of f(x) = |x| to the graph of g(x) = |4+x|
A. Vertical translation of 4 units down
B. Horizontal translation of 4 units left
C. Vertical translation of 4 units up
D. Horizontal translation of 4 units right
Answers
Adding a number to the variable moves the graph horizontally to the left.
The answer is B.
there are 230 books in a library and 60% of them are fiction. how many fictional bools are there
Answers
To find the number of fictional books in the library, multiply the total number of books by 0.60 which is 138.
Explanation:To find the number of fictional books in the library, we need to calculate 60% of the total number of books. Thus the concept of percentage calculation will be followed.
Calculating 60% of 230 -
= 60/100 x 230
= 13800/100
= 138
Therefore, there are 138 fictional books in the library.
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Which list shows the numbers in order from least to greatest?
2.43, 2.045, 259
2.045, 2.43, 259
2.43, 2.045, 259
259 , 2.045, 2.43
2.045, 259 , 2.43
Answers
Answer: second one (B) - 2.045, 2.43, 259
Step-by-step explanation: 2.045 is the lowest value due to the 0 in the hundredths place. 2.43 has a lower value than 259, it being a decimal.
Marie placed two orders at the same restaurant last month. In one order, she received 5 trays of lasagna and 3 trays of garlic bread for $163. In the other order, she received 4 trays of lasagna and 4 trays of garlic bread for $150.80. What is the price for one tray of lasagna, l, and one tray of garlic bread, g? l = $24.95 g = $12.75 l = $12.75 g = $24.95 l = $24.75 g = $12.95 l = $12.95 g = $24.75
Answers
Answer:
L=$24.95 , G=$12.75
Step-by-step explanation:
This problem is a classic system of equations problem. There are several methods of solving these problems, because of the circumstances and values given I think it's best to use the elimination method. The elimination method attempts to cancel out a variable by subtracting one equation from another.
The first statement Marie received an order of 5 trays of lasagna and 3 trays of garlic bread for a total of $163. I'm going to use "L" for lasagna and "G" for garlic bread. Therefore, the equation for the scenario above is:
[tex]5L+3G=163[/tex]
In the second statement Marie received an order of 4 trays of lasagna and 4 trays of garlic bread and so:
[tex]4L+4G=150.80[/tex]
Now we cannot use the elimination method directly because equation 1 and equation 2 do not have any coefficients in common. However, we can eliminate "G" by first multiplying the top equation by 4 and the bottom equation by 3. We are doing this because those are the coefficients of the other equation, and thus will return a common value as such:
Equation 1:
[tex]4(5L+3G=163)\\20L+12G=652[/tex]
Equation 2:
[tex]3(4L+4G=150.80)\\12L+12G=452.40[/tex]
Now we can subtract equation 2 from equation 1 and eliminate the "G" variable as such:
[tex]20L+12G=652\\-(12L+12G=452.40)\\\\8L+0G=199.60\\\\8L=199.60\\\\[/tex]
By solving for L we obtain:
[tex]8L=199.60\\\\\frac{8L}{8}=\frac{199.60}{8}\\\\L=24.95[/tex]
Therefore, one tray of lasagna is $24.95. Now that we know the value of "L" we can use any of our original equations to figure out what "G" is. I'll use the first equation and so:
[tex]5L+3G=163\\\\5(24.95)+3G=163\\\\124.75+3G=163\\\\124.75-124.75+3G=163-124.75\\\\3G=38.25\\\\\frac{3G}{3}=\frac{38.25}{3}\\\\G=12.75[/tex]
And so the price of a tray of garlic bread is $12.75. So the answer is the price of lasagna = $24.95 and garlic bread = $12.75
~~~Brainliest Appriciated~~~
Answer:
lasagna = $24.95 and garlic bread = $12.75
:)
2. Marco has a jug with 37 ½ ounces or juice. He wants to use some small cups to pass out samples of the juice. Each cup hold 2 ½ ounces of juice. How many cups can marco fill with juice samples?
Answers
Answer: Marco can fill 15 of the 2 1/2 Cups
Step-by-step explanation:
37.5 divided by 2.5
or
37 1/2 divided by 2 1/2
Answer:
Marco can fill 15 of the 2 1/2 Cups
37 1/2 divided by 2 1/2
A trail mix that costs $2.45 per pound is mixed with a trail mix that costs $2.30 per pound. How much of each type of trail mix must be used to have 30 pounds of trail mix that costs $2.35 per pound?
Answers
Answer:
10 pounds of trial mix that costs $2.45 and 20 pounds of trial mix that costs $2.30
Step-by-step explanation:
[tex]\\\text{Let x pounds of the trial mix that costs \$2.45 per pound, and y pounds of the trial mix}\\\text{that costs \$2.30 per pound are added to get a 30 pounds of trial mix }\\\text{that costs \$2.35 per pound. since the total mixture is 30 pounds, so we get}\\\\x+y=30\ \ \ \ \ \ \ \ \ \ \ \ \ ...... (i)\\\\\text{and the total cost equation is given by}\\\\\$2.45 x + \$2.30 y=\$2.35 (30)\\\\2.45 x + 2.30 y=70.5 \ \ \ \ \ \ \ \ \ \ \ \ \ ...... (ii)\\[/tex]
[tex]\text{now to solve the equation (i) and (ii), we'll substitute }y=30-x \text{ from (i) into (ii)}\\\text{so we get}\\\\2.45x+2.30(30-x)=70.5\\\\2.45x+69-2.30x=70.5\\\\\Rightarrow 2.45x-2.30x=70.5-69\\\\\Rightarrow 0.15x=1.5\\\\\Rightarrow x=\frac{1.5}{0.15}\\\\\Rightarrow x=10\\\\\text{plug this value of x in (i), we get}\\\\y=30-x=30-10=20\\\\\text{so we must add 10 Pounds of trial mix that costs \$ 2.45 and 20 pounds of}\\\text{the trial mix that costs \$2.30}[/tex]
Final answer:
To create a 30-pound trail mix costing $2.35 per pound, mix 20 pounds of $2.45 trail mix with 10 pounds of $2.30 trail mix.
Explanation:
Trail mix costing: $2.45 per pound and $2.30 per pound must be mixed to get a 30-pound mix costing $2.35 per pound.
To calculate:
Let x = pounds of $2.45 mix and y = pounds of $2.30 mix.
Set up the system of equations: x + y = 30 and (2.45x + 2.30y)/30 = 2.35.
Solve the system to find x = 20 pounds of $2.45 mix and y = 10 pounds of $2.30 mix.
Sarah buys a video game for $44.00 plus 4% sales tax. Her brother, Jack, buys a video game for $36.00 plus 6% tax. Show your work. (3 points)
a.) What is the total cost of Sarah’s video game? Show all work.
b.) What is the total cost of Jack’s video game? Show all work.
c.) Who spent more on tax?
Answers
Answer:
Sarah spent 45.76
Jack spent 38.16
Jack spent more on tax
Step-by-step explanation:
Sarahs game costs:
tax = 44 *.04 = 1.76
Total cost = game + tax
= 44 + 1.76
=45.76
Jacks game costs:
tax = 36 *.06 = 2.16
Total cost = game + tax
= 36 + 2.16
=38.16
Jack spent more on tax
Bryant Peters bought 75 shares of Elkston stock for $24.13 per share. The company paid annual dividends of $0.29 per share. What is the total annual dividend? Show work.
A) 32.10
B) 21.99
C) 21.75
D) 22.00
Answers
Answer:
The right option is C) 21.75
Step-by-step explanation:
we have given,
Number of shares = 75
Cost of each share = $24.13
Total cost of shares = 75×24.13 = $1809.75
since the company paid annual dividends of $0.29 per share.
Total annual dividends company paid = 75×0.29 = $21.75
Hence the right option is C) 21.75
Find f(x) and g(x) so that the function can be described as y=f(g(x))
y=(9/x^2)+2
Answers
there are many combinations for it, but we can settle for say
[tex]\bf \begin{cases} f(x)=x+2\\[1em] g(x)=\cfrac{9}{x^2}\\[-0.5em] \hrulefill\\ (f\circ g)(x)\implies f(~~g(x)~~) \end{cases}\qquad \qquad f(~~g(x)~~)=[g(x)]+2 \\\\\\ f(~~g(x)~~)=\left[ \cfrac{9}{x^2} \right]+2\implies f(~~g(x)~~)=\cfrac{9}{x^2}+2[/tex]
plz help Pru, Piper, and Phoebe share spices in the ratio 5:3:4. If Pru has 24 more spices than Piper, how many spices does Phoebe have?
Answers
Answer:
48 spices
Step-by-step explanation:
The number of spices each has is a multiple of 5, 3, and 4.
The number of spices each has is 5x, 3x, and 4x.
Pru has 5x spices.
Piper has 3x spices.
Pru has 24 more spices than Piper, so
5x = 3x + 24
2x = 24
x = 12
Phoebe has 4x spices, so Phoebe has
4(12) = 48
So remember that ratios are part to part, not part to whole like fractions are. In this case:
5 + 3 + 4 = 12
There are 12 parts total. This means that Pru has 5/12 of the spices, Piper has 3/12 (or 1/4) of the spices and Phoebe has 4/12 (or 1/3) of the spices. Now, we also know that Pru has 24 more spices than piper. With this info, we can set up our equation as (x = total amount of spices):
[tex]\frac{5}{12}x=\frac{3}{12}x+24[/tex]
From here we can solve for the total amount of spices. Firstly, subtract both sides by 3/12x:
[tex]\frac{2}{12}x=24[/tex]
Next, multiply both sides by 12/2 (or 6):
[tex]x=144[/tex]
There are a total of 144 spices. Now, to get Phoebe's amount of spices multiply 1/3 by 144:
[tex]\frac{144}{1}*\frac{1}{3}=\frac{144}{3}=48[/tex]
Answer:In short, Phoebe has 48 spices.
What is the value of the problem below?
Answers
Answer:
90
Step-by-step explanation:
The first term = 6* 2^0 = 6
The second term = 6 * 2^ (2-1) = 6*2 = 12
The third term = 6* 2^(3-1) = 6*2^2 = 6*4 = 24
The fourth term = 6* 2^(4-1) = 6* 2^3 = 6*8 = 48
S4 is the sum of the 1st four terms
S4 = 6+ 12+24+ 48 = 90
Find X and K in this pair of similar triangles.
Answers
Answer:
side X is 5 and side k is 6
Step-by-step explanation: