If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally.
The Triangle Proportionality Theorem asserts that a line drawn parallel to one side of a triangle divides the other two sides proportionally. This is related to the properties of similar triangles, where the corresponding sides of similar triangles are proportional due to the AAA similarity criterion.
The Triangle Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. This means that the segments created on one side of the triangle are in the same ratio as the corresponding segments on the other side of the triangle. For example, in [tex](\Delta ABC)[/tex] with a line parallel to [tex](\overline{BC})[/tex], the theorem asserts that [tex](\frac{AD}{DB} = \frac{AE}{EC})[/tex].
This concept is deeply rooted in the properties of similar triangles which are based on the Angle-Angle-Angle (AAA) similarity criterion. According to the AAA theorem, if two triangles have corresponding angles that are equal, then their corresponding sides are proportional. Thus, if two triangles are similar, the lengths of their corresponding sides form a proportion such as [tex](\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF})[/tex], where each fraction is the ratio of a pair of corresponding sides.
The principles of the Triangle Proportionality Theorem can be applied in various geometric problems, often to compare the lengths of sides within the same triangle or in two similar triangles. It is also useful in determining distances or heights by applying the theorem to real-life scenarios, such as calculating the height of a wall using the shadow it casts, provided the sun's rays are considered parallel.
Some penguins were playing on the ice. Ten of them dove in the water. A few seconds later, 6 of those penguins jumped back onto the ice. Now there are 12 penguins on the ice. How many penguins were on the ice to start? Use numbers, pictures,or words to show how you got the answer.
There were 16 penguins on the ice at the start.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let x represent the amount of penguins at start, hence:
x - 10 + 6 = 12
x - 4 = 12
Add 4 to both sides of the equation:
x - 4 + 4 = 12 + 4
x = 16
There were 16 penguins on the ice at the start.
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The height of a triangle is twice the length of its base. The area of the triangle is 50m^2. Find the height and base to the nearest tenth of a meter
Plzz help me asap i need help
I willgive brainliest
which shows all factors pairs of 45.
A line passes through the points (–5, 2) and (10, –1). Which is the equation of the line?
Answer:
Y=-1/5x + 1
Step-by-step explanation:
the gas station is 44 kilometers away. how far is the station in miles if 1 mile is 1.6 kilometers
A 10-ft-long slide is attached to a deck that is 5 ft high. Find the distance from the bottom of the deck to the bottom of the slide to the nearest tenth.
Final answer:
The distance from the bottom of the deck to the bottom of the slide is approximately 8.7 feet when calculated using the Pythagorean theorem for a right-angled triangle formed by the slide, the deck, and the ground.
Explanation:
The student wants to find the horizontal distance from the base of the deck to the bottom of a 10-ft-long slide that makes a right angle with the deck which is 5 ft high. This scenario forms a right-angled triangle where the slide is the hypotenuse, the deck is one leg (height), and the distance from the base of the deck to the bottom of the slide is the other leg (base).
To solve for the distance (d), which is the base of the triangle, we can use the Pythagorean theorem:
a² + b² = c²
where
a is the height of the deck (5 ft)b is the distance from the bottom of the deck to the end of the slide (d)c is the length of the slide (10 ft)Plugging the known values into the equation, we get:
5² + d² = 10²
25 + d² = 100
d² = 100 - 25
d² = 75
d = √75
d ≈ 8.7 ft (to the nearest tenth)
Whats the distance from point y to qx in the figure
Owen’s math book weighs 2 pounds 13 ounces. His science book weighs 1 pound 15 ounces. His backpack weighs 1 pound 1 ounce. What is the total weight in ounces of the backpack and two books, please explain how you got the answer.
Will mark as Brainliest.
The number of wild flowers growing each year in a meadow is modeled by the function f(x).
f(x)= \frac{1000}{1+9e^{-0.4x} }
Which statements are true about the population of wild flowers?
Select each correct answer.
A. Initially there were 100 wild flowers growing in the meadow.
B. 42 more wildflowers will grow in the 11th year than in the 10th year.
C. After approximately 9 years, the rate for the number of wild flowers decreases.
D. In the 15th year, there will be 1050 wild flowers in the meadow.
Using the provided function, statement A is true as there were initially 100 wildflowers. Statements B, C, and D require calculations using the function and its derivative to determine their truth, which has not been provided in this format.
The function f(x) = \frac{1000}{1+9e^{-0.4x}} models the number of wild flowers growing each year in a meadow. Let's evaluate the truth of the given statements.
A. Initially there were 100 wild flowers: To verify, we calculate f(0). Since e^0 is 1, f(0) = \frac{1000}{1+9(1)} = \frac{1000}{10} = 100. This statement is true.B. 42 more wildflowers in the 11th year than in the 10th year: We need to calculate f(11) - f(10). If this difference is 42, then the statement is true. After calculations, if the result is not 42, then the statement is false.C. After approximately 9 years, the rate of growth decreases: We analyze the behavior of the function's derivative to determine the changing rates. If the derivative decreases after 9 years, this statement is true.D. In the 15th year, there will be 1050 wild flowers: We calculate f(15). If it equals 1050, then the statement is true. If not, it is false.To answer these, we would need to calculate values explicitly using the function provided, and for statement C, we would need to find the first derivative and analyze its behavior over time.
Solve the linear equation 2.25 – 11j – 7.75 + 1.5j = 0.5j – 1.
j = –0.45
j = –0.25
j = 0.25
j = 0.45
Answer:
Based on calculations the correct answer is A
Step-by-step explanation:
A bicycle goes on sale at 75% of its original price of $160. What is its sale price?
How do you write 5/9 as a percentage
Answer:
55.5
Step-by-step explanation:
divide numerator by denominator
Maya’s bedroom is in the shape of a rectangular prism 15 feet long 12 feet wide and 10 feet high it has no windows Maya once to P allfour walls including the door in your ceiling where is the service area she will paint
Plz Help Meee!!!!! >.<
Jessica is baking a cake. The recipe says that she has to mix 32 grams of vanilla powder to the flour. Jessica knows that 1 cup of vanilla powder has a mass of 128 grams. She added ¼ cup of vanilla to the flour. Should she add more vanilla powder to make the exact recipe, did she go over, or did she use the perfect amount? How do you know?
Jessica added exactly ¼ cup of vanilla powder, weighing 32 grams, to her recipe. This is the perfect amount as stated by the cake recipe.
Explanation:Jessica is tasked with following a cake recipe that requires 32 grams of vanilla powder. She knows that 1 cup of vanilla powder weighs 128 grams. Since she added only ¼ cup of vanilla powder, we need to calculate if this amount is equivalent to the required 32 grams.
First, let's determine how many grams are in ¼ cup of vanilla powder:
Since 1 cup is 128 grams, ¼ of that amount is:Therefore, Jessica has added the perfect amount of vanilla powder to her recipe.
a certain number is the same as the square root of the product of 8 and the number find the number
The number is given by the equation n = 8
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
Let the number be represented as n
And , a number is the same as the square root of the product of 8 and the number
Substituting the values in the equation , we get
n = √ ( 8 x n ) be equation (1)
On simplifying the equation , we get
n = √8n
Taking squares on both sides of the equation , we get
n² = 8n
Divide by n on both sides of the equation , we get
n = 8
Therefore , the number is 8
Hence , the equation is n = 8
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how do you convert degrees to radians?
One degree is equal 0.01745329252 radians:
1° = π/180° = 0.005555556π = 0.01745329252 rad
The angle α in radians is equal to the angle α in degrees times pi constant divided by 180 degrees:
α(radians) = α(degrees) × π / 180°
or
radians = degrees × π / 180°
ExampleConvert 30 degrees angle to radians:
α(radians) = α(degrees) × π / 180° = 30° × 3.1459 / 180° = 0.5236 rad
Degrees to radians in terms of piThe angle α in radians is equal to the angle α in degrees times pi constant divided by 180 degrees:
α(radians) = (α(degrees) / 180°) × π
ExampleConvert 30 degrees angle to radians in terms of pi:
α(radians) = (α(degrees) / 180°) × π = (30° / 180°) × π
= (1/6) × π = π/6 rad = 0.166667π rad
Final answer:
To convert degrees to radians, multiply the number of degrees by π and then divide by 180, or simply multiply by π/180. For instance, 45 degrees can be converted to radians as 45 degrees * (π/180) = π/4 radians.
Explanation:
To convert degrees to radians, you can use the basic formula that π radians equal 180 degrees. This means that to convert a degree measure to radians, you multiply the number of degrees by π and then divide by 180. This can be simplified to multiplying the number of degrees by π/180.
For example, if you wanted to convert 45 degrees to radians, you would perform the following calculation:
45 degrees * (π/180) = 45π/180 = π/4 radians.That means 45 degrees is equivalent to π/4 radians. Remember, the π symbol represents the mathematical constant pi, which is approximately 3.14159.
Please help me
i gotta turn this in in 10 minutes
Different types of parallelograms have distinguishing traits. Compare and contrast the rectangle, rhombus, and square based upon the characteristics of their sides. Which one(s) have both pairs of opposite sides that are parallel? Which one(s) have both pairs of opposite sides that are congruent? Which one(s) have all four sides that are congruent?
What is the value of x (10x-20) (6x+8)
Answer:
7.
Step-by-step explanation:
2 1/4 simplified with 1/4
2 1/4 +1/4
9/4+1/4=10/4=5/2=1 3/2
1 3/2
whats the slope of the line y+6=1/3(x-4)
There are 14 children in Erica's preschool class. During free time yesterday, 2 of them chose to do puzzles. What is the probability that a randomly selected child chose to do puzzles yesterday?
1/7 or 2/14... it is the same thing... :-)
I’m confused on this question.
Please help.
The answer is $30,000.00
The way you find this is by using the formula, Percent x Part=Whole
.15 or 15% x $45,000.00= $30,000.00
hopes this helps,
Xeno
Which function is shown in the graph?
And a three week vacation to Paris Martha's expenses on food gifts and accommodations was $80 less than three times for airfare if the total expenses for the trip was $2660 how much was your airfare?
PLEASE HELP!!
1. calculate the length of the track
2. how many laps do you need to travel 1609 meters (about 1 mile)?
3. how much of a rectangular area does the track take up
Note: turns 1, 2, 4, 5, 6, 8 and 9 all have a radius of 3 meters. Turns 3 and 7 each have a radius of 2.25 meters
#1) 163.81 m
#2) About 9.8 laps.
#3) 1314 square meters
Explanation
#1) We first add all of the horizontal and vertical sections:
30+8+6+6+12+30+20 = 112 m
Now we find the distance around each curve. The distance around a circle is the circumference; the formula for circumference is
C=πd.
We will use 3.14 for π for each curve:
Curve 5: The radius is 3, so the diameter is 2*3 = 6. This is 1/4 of a curve, so we multiply the circumference by 1/4:
1/4(3.14)(6) = 4.71
Curve 4: This is another 1/4 curve with a radius of 3, so the distance around is 4.71.
Curve 3: This is 1/2 of a curve, so we will multiply the circumference by 1/2. The radius is 2.25, so the diameter is 2*2.25 = 4.5:
1/2(3.14)(4.5) = 7.065
Curve 2: This is another 1/4 curve with a radius of 3, so the distance around is 4.71.
Curve 1: This is another 1/4 curve with a radius of 3, so the distance around is 4.71.
Curve 9: This is another 1/4 curve with a radius of 3, so the distance around is 4.71.
Curve 8: This is another 1/4 curve with a radius of 3, so the distance around is 4.71.
Curve 7: This is another 1/2 curve with a radius of 2.25, so the distance around is 7.065, just as curve 3.
Curve 6: This is 1/2 of a curve with a radius of 3, so the distance around is 1/2(3.14)(2*3) = 9.42
The total distance around the curved parts is:
4.71+4.71+7.065+4.71+4.71+4.71+4.71+7.065+9.42 = 51.81
Together this gives us 112+51.81 = 163.81.
#2) To find the number of laps it takes to go 1609 meters, divide:
1609/163.81 = 9.8
#3) To find the rectangular area, we first need the total distance across and the total distance vertically.
Horizontally, we have a radius of 3, a horizontal distance of 30, and another radius of 3:
3+30+3 = 36
(looking at the right hand side)
Vertically, we have a radius of 3, vertical distance of 8, radius of 3, radius of 2.25, radius of 2.25, radius of 3, vertical distance of 12, and radius of 3:
3+8+3+2.25+2.25+3+12+3 = 36.5
The area is then 36(36.5) = 1314 m²
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Which number is not a perfect square?
A) 4
B) 9
C) 144
D) 84
Jamie’s dad put 12 gallons of gasoline in his car the gasoline costs 3.25$ a gallon how much did Jamie’s dad pay for the gasoline
To find out how much Jamie's dad paid for the gasoline, we need to multiply the amount of gasoline he bought by the cost per gallon.
We know that Jamie's dad bought 12 gallons of gasoline, and the cost per gallon is $3.25.
So, let's multiply these two numbers together.
First, take the 12 gallons of gasoline.
Then, multiply that number by the cost per gallon, which is $3.25.
So, 12 × $3.25 equals $39.00.
Therefore, Jamie's dad paid for the gasoline.
answer: $39.00