Answer:
the graph of the parabola opens upwards.
Step-by-step explanation:
For any quadratic equation of the form
[tex]ax ^ 2 + bx + c[/tex] is true that:
if the main coefficient "a" is negative then the graph of the parabola opens downwards.
If the main coefficient "a" is positive, the parabola opens upwards
In this case the parabola is [tex]y=x^2-6x[/tex]
Note that [tex]a=1[/tex] and [tex]a>0[/tex] therefore the graph of the parabola opens upwards.
Lewis has a bag of colored marbles. The bag contains 24 red marbles , 36 blue marbles , and 60 yellow marbles. What are the ratios of the number of red marbles , blue marbles , and yellow marbles to the total number of marbles?
Answer:
Step-by-step explanation:
The total number of marbles is 24 + 36 + 60 = 120
Red: 24/120 = 1/5
Blue: 36/120 = 6/20 = 3/10
Yellow: 60 / 120 = 1/2
Factor completely and find the roots of the following. X^2+12x+27=0
Answer:
x = - 9, x = - 3
Step-by-step explanation:
Given
x² + 12x + 27 = 0
To factorise the quadratic
Consider the factors of the constant term (+ 27) which sum to give the coefficient of the x- term (+ 12)
The factors are + 9 and + 3, since
9 × 3 = 27 and 9 + 3 = 12, hence
(x + 9)(x + 3) = 0
Equate each factor to zero and solve for x
x + 9 = 0 ⇒ x = - 9
x + 3 = 0 ⇒ x = - 3
What is the value for f(x) = 42x - 100 when x= 2 ?
Final answer:
To find the value of f(x) when x=2 for the function f(x) = 42x - 100, substitute 2 for x, giving f(2) = 84 - 100, which simplifies to f(2) = -16.
Explanation:
The student is asking to evaluate the function f(x) = 42x - 100 when x = 2. To do this, we simply substitute 2 for x in the equation, which gives us f(2) = 42(2) - 100. Performing the multiplication first, we get 84 - 100. Thus, f(2) equals -16.
The subject is mathematics, specifically related to the evaluation of functions, which is a topic typically covered in high school algebra courses. This is a straightforward example of how to evaluate a function at a given point.
Find the value of x to the nearest tenth.
Answer:
x = 8.9 (nearest tenth)
Step-by-step explanation:
6^2 - 4^2 = 36 - 16 = 20
So
x = 2 * (√20)
x = 2 * 4.47
x = 8.94
Answer:
The value of x = 4√5
Step-by-step explanation:
Points to remember
For a right angled triangle
Hypotenuse² = Base² + Height²
To find the value of x
From the figure we can see a right angled triangle with,
hypotenuse = 6 and height = 4
Value of x = 2 * base
we have, Hypotenuse² = + Height²
Base² = Hypotenuse² - Height²
= 6² - 4²
= 36 - 16 = 20
Base = √20 = 2√5
x = 2 * 2√5 = 4√5
The value of x = 4√5
A basket contains 6 oranges and 4 tangerines. a sample of 3 is drawn. find the probability that they are all oranges
The probability of drawing 3 oranges from a basket of 6 oranges and 4 tangerines is 1/6 or approximately 0.167.
Explanation:This problem is a probability question in the field of mathematics. It is asking us to find the probability that all three fruits drawn from the basket are oranges. This can be solved using the concept of combinations from probability theory.
The total number of fruits in the basket is 10 (6 oranges and 4 tangerines). We want to draw a sample of 3 fruits. The number of ways to draw 3 fruits from a total of 10 (regardless of type) is given by the combination formula C(n, r), where n is the total number of items and r is the number of items to choose. So we have C(10, 3) = 120 possible ways.
Next, consider the scenario we're interested in--drawing 3 oranges. The total number of ways to draw 3 oranges from a total of 6 oranges is C(6, 3) = 20 possible ways.
So, the probability of drawing 3 oranges is the number of ways to draw 3 oranges divided by the total number of ways to draw 3 fruits. So, the probability is 20/120 = 1/6 or approximately 0.167.
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Final answer:
The probability of drawing 3 oranges from a basket containing 6 oranges and 4 tangerines is found using combinations. The calculation shows that the probability of selecting all oranges in a draw of 3 fruits is 1/6.
Explanation:
The question asks to find the probability that when three fruits are drawn from a basket containing 6 oranges and 4 tangerines, all three are oranges. This is a problem of probability without replacement, which can be solved using combinations.
To find the probability of drawing 3 oranges out of 3 fruits, we calculate the combinations of choosing 3 oranges from 6 and then divide it by the combinations of choosing 3 fruits from the total 10 fruits in the basket.
The number of ways to select 3 oranges out of 6 is given by the combination formula C(n, k) = n! / (k!(n - k)!), where n is the total number of items, and k is the number of items to choose. So, we have:
Ways to pick 3 oranges: C(6, 3)
Total ways to pick 3 fruits: C(10, 3)
Then the probability is calculated by:
P(all oranges) = C(6, 3) / C(10, 3)
Calculating the combinations, we have:
P(all oranges) = (6! / (3! × (6-3)!)) / (10! / (3! × (10-3)!)) = (6 × 5 × 4) / (3 × 2 × 1) divided by (10 × 9 × 8) / (3 × 2 × 1) = 20 / 120 = 1/6.Thus, the probability that all three fruits are oranges is 1/6.
20% of 180ft what is the quantity
Answer: 36ft
Step-by-step explanation: 180*.2=36
Answer:
36 ft
Step-by-step explanation:
to find 20% of 180 ft, multiply 180 ft by 0.20: 0.20(180 ft) = 36 ft
9 please help with 9
10 what is P
Answer:
9.
The probabilities are the same
10.
1/3
Step-by-step explanation:
9.
Assuming the coin is fair;
The probability of getting heads in a single toss is, P(H) = 1/2
The probability of getting tails in a single toss is, P(T) = 1/2
Now, P(H,H,H) is the probability of obtaining 3 heads in the e tosses. Since each toss is independent of the others,;
P(H,H,H) = P(H)*P(H)*P(H)
= 1/2 * 1/2 * 1/2 = 1/8
On the other hand, P(H,T,H) is the probability of obtaining a head followed by a tail followed by a final head. Using the independence property;
P(H,T,H) = P(H)*P(T)*P(H)
= 1/2 * 1/2 * 1/2 = 1/8
Therefore, P(H,H,H) = P(H,T,H)
10.
We are informed that a coin is tossed and a number cube is rolled. We are to determine the following probability;
P(heads, a number less than 5)
Assuming the coin is fair;
The probability of getting heads in a single toss is, P(H) = 1/2
Assuming the number cube is fair as well, the probability of rolling a number less than 5 is;
4/6 = 2/3
This is because there are 4 numbers less than 5, (1, 2, 3, 4) while the cube has 6 sides.
P(heads, a number less than 5), the events presented here are independent since the outcome from tossing the coin does not in any way determine the outcome from rolling the number cube. Therefore,;
P(heads, a number less than 5) = P(heads)*P(a number less than 5)
= 1/2 * 2/3 = 1/3
The table below shows 10 data values:
224 245 203 290 217
236 262 216 224 245
What values of minimum, Q1, median, Q3, and maximum should be used to make a box plot for this data?
Minimum = 203, Q1 = 245, median = 230, Q3 = 217, maximum = 290
Minimum = 203, Q1 =217, median = 230, Q3 = 245, maximum = 290
Minimum = 216, Q1 = 236, median = 240, Q3 = 245, maximum = 262
Minimum = 216, Q1 =245, median = 240, Q3 = 236, maximum = 262
Answer:
The answer is the second option
Minimum = 203, Q1 =217, median = 230, Q3 = 245, maximum = 290
Step-by-step explanation:
First order the data from lowest to highest
203, 216, 217, 224, 224, 236, 245, 245 ,262 ,290
Minimun = 203
Maximum =290
Now divide the data in half
203, 216, 217, 224, 224, 236, 245, 245 ,262 ,290
Divide half of the first half of the data. The data of the medium is the first quartile
1) 203, 216, [217], 224, 224
Q1
Divide half of the second half of the data. The data of the medium is the third quartile
2) 236, 245, [245] ,262 ,290
Q3
The average between the two values that are in the center of the 10 ordered data is the median
203, 216, 217, 224, [224, 236], 245, 245 ,262 ,290
[tex]Median =\frac{224+236}{2}=230[/tex]
Median=230
The answer is the second option
Minimum = 203, Q1 =217, median = 230, Q3 = 245, maximum = 290
Answer:
the answer is B :D
Step-by-step explanation:
plz help and god bless
What is the median of Restaurant A's cleanliness ratings?
1
2
3
4
5
Answer: 3
Step-by-step explanation:
Median is the middle number if it is an odd number or if it is even you add the two middle numbers and divide by 2
The median cleanliness rating for Restaurant A is 3.
Explanation:The median of Restaurant A's cleanliness ratings can be found by arranging the ratings in order from lowest to highest and determining the middle value. In this case, the ratings are 1, 2, 3, 4, and 5. Since there is an odd number of ratings, the median is the middle value, which is 3. Therefore, the median cleanliness rating for Restaurant A is 3.
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What is the radius of each semicircle in the following composite figure?
https://isd402.owschools.com/media/g_mat07_2016/9/img_testa2_composite_figure.gif
9 yd
6 yd
4.5 yd
3 yd
Answer: 3 yd
Step-by-step explanation: The radius is 3 yards because the diameter is 6 and the radius is d/2. 6/2 is equal to 3.
Hope this helps!
What is 7x1.5gallons
Answer: 10.5 gallons
Step-by-step explanation:
One imperial gallon is approximately 1.2 US gallons. The US gallon is used in the United States and is equal to exactly 231 cubic inches or 3.785411784 liters. The Imperial gallon or UK gallon is used in the United Kingdom and is equal to approximately 277.42 cubic inches. Its exact value is defined as 4.54609 liters.
I have here $12 would i divide it by 1/3 or divide it by 1/4
Answer:
You could divide $12 with either method
If you divided 12 by 1/4 you would get $48.
Divided by 1/3, $36.
How do you figure this out?
Answer:
∠g and ∠h are complementary angles∠g and ∠h are acute anglesStep-by-step explanation:
Use the given information to determine what the angles can be.
g = 2x -90 . . . . given
g > 0 . . . . . . . . . given
2x -90 > 0
2x > 90 . . . . . add 90
x > 45 . . . . . . divide by 2
__
h = 180 -2x
h > 0
180 -2x > 0
180 > 2x
90 > x
__
The requirement that both angles be greater than zero puts limits on x:
45 < x < 90
We can put this back into the given relations for g and h:
g = 2x -90
x = (g +90)/2
45 < (g +90)/2 < 90 . . . . substitute for x
0 < g/2 < 45 . . . . . . . . . . subtract 45
0 < g < 90 . . . . . . . . . . . . g is an acute angle
Similarly, ...
h = 180 -2x
x = (180 -h)/2 = 90 -h/2
45 < (90 -h/2) < 90 . . . . substitute for x
-45 < -h/2 < 0 . . . . . . . . . subtract 90
90 > h > 0 . . . . . . . . . . . multiply by -2; h is an acute angle
__
We can add the angle measures to see if they are supplementary or complementary:
g + h = (2x -90) +(180 -2x)
g + h = 90 . . . . . simplify; the angles are complementary
__
The relevant observations are ...
∠g and ∠h are complementary angles∠g and ∠h are acute anglesUsing the numbers 8, 6, 4, and 2 write an expression that equals 40.
Answer:
[tex]\large\boxed{(8\times6)-(4\times2)=48-8=40}[/tex]
The expression (8*2)*2 + 6 + 2 uses the numbers 8, 6, 4, and 2 to equal 40. The question tests knowledge of basic arithmetic operations.
Explanation:The question involves using the numbers 8, 6, 4, and 2 to create an expression that equals 40. This is a problem dealing with basic arithmetic operations like addition, subtraction, multiplication, and division. The expression can be formed as follows:
Multiply 8 by 2. (8*2 = 16). Multiply 16 by 2. (16*2 = 32). Add 6 to 32. (32+6 = 38). Add 2 to the 38. (38+2 = 40).
So, the expression is: (8*2)*2 + 6 + 2 = 40
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2.
The quadrilateral shown below has vertices at (-8, 0).(-4,-4), (0,8), and (4,4).
What is the area of the quadrilateral?
The area is 64. Area, as you probably know, is the length times the width of the figure. The length from A to C is approximately radical 128, and the length from C to D is approximately radical 32 (btw, the radical is the check mark with line that goes above and next to the radicand (the number on the inside of the radical)). Multiply these two to get the area, and you should end up with 64.
If the quadrilateral has vertices at (-8, 0), (-4,-4), (0,8), and (4,4). Then the area of the quadrilateral will be 64 square units.
What is a quadrilateral?It is a polygon with four sides. The total interior angle is 360 degrees.
The quadrilateral shown below has vertices at (-8, 0), (-4,-4), (0,8), and (4,4).
Then the area of the quadrilateral will be
Assume the points
(x₁, y₁) = (-8, 0)
(x₂, y₂) = (-4, -4)
(x₃, y₃) = (4, 4)
(x₄, y₄) = (0, 8)
Then the area of the quadrilateral is given as
Area = 1/2 |[(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁)]|
Area = 1/2|[(-8 × -4 + -4×4 + 4×8 + 0×0) - (0 × -4 + -4 × 4 + 4 × 0 + 8 × -8)]|
Area = 1/2|[(32 - 16 + 32) - ( -16 - 64)]|
Area = 1/2|[48 + 80]|
Area = 1/2|128|
Area = 64
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What is the measure of mrn
16
36
71
87
Answer:
16
Step-by-step explanation:
Exterior angle theorem - the sum of two interior angles (which in this case would be angle R and N) is equal to the measure of the exterior angle.
One city has a population of about 11,566,740. Another city has a population of about 6,978,730. What would the combined population be?
Answer:
Step-by-step explanation:
11,566,740
+6,978,730
--------------------
18545470
Not much explaining, just add. A calculator can be used.
The combined population of the city will be 18,545,470.
What is addition?Addition is the mathematical operation of adding two numbers to get total.
We have,
Population of One city = 11,566,740
And,
Population of another city = 6,978,730
Now,
To get combined population,
Add population of both city,
i.e.
Combined population = Population of One city
+ Population of another city
= 11,566,740 + 6,978,730
i.e.
Combined population = 18,545,470
Hence, we can say that the combined population of the city will be 18,545,470.
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2 Points
What is the slope of the line shown below
(3,9)
(1,1)
ANSWER
The slope is 4
EXPLANATION
We want to find the slope of the line goin through (3,9) and (1,1)
We use the slope formula:
[tex]m = \frac{y_2-y_1}{x_2-x_1} [/tex]
We substitute the points into the slope formula to get;
[tex]m = \frac{1 - 9}{1 - 3} [/tex]
This simplifies to:
[tex]m = \frac{ - 8}{ - 2} [/tex]
[tex]m = 4[/tex]
Hence the slope of the line is 4.
Given: ΔPSQ, PS = SQ PΔPSQ = 50 SQ – PQ = 1 Find: Area of ΔPSQ
Answer:
120 square units
Step-by-step explanation:
In triangle PSQ, PS=SQ. Let PS=SQ=x units.
Since SQ-PQ=1, PQ=SQ-1=x-1 units.
The perimeter of the triangle PSQ is 50 units, so
PS+SQ+PQ=50 units.
Substitute PS=SQ=x un. and PQ=x-1 un.
x+x+x-1=50
3x=51
x=17
Hence
PS=SQ=17 units,
PQ=16 units.
Use Heron's formula to find the area:
[tex]A=\sqrt{p(p-a)(p-b)(p-c)},[/tex]
where p is semi-perimeter and a,b,c are lengths of sides.
[tex]p=\dfrac{17+17+16}{2}=25,\\ \\\\A=\sqrt{25(25-17)(25-17)(25-16)}=\sqrt{25\cdot 8\cdot 8\cdot 9}=5\cdot 8\cdot 3=120\ un^2.[/tex]
What is the volume of this prism?
units3=
Answer:
240 units^3
Step-by-step explanation:
To find the volume of a rectangular prism you just multiply length times width times height. So, 8*6*5 which equals 240 units^3.
The dimensions of the prism is given as Length = 8 units, width = 6, Height = 5. Therefore, the volume of the prism is 240 units^3.
How to find the volume of a right rectangular prism?Suppose that the right rectangular prism in consideration be having its dimensions as 'a' units, 'b' units, and 'c' units,
then its volume is given as:
[tex]V = a\times b \times c \: \: unit^3[/tex]
The dimensions of the prism is given as
Length = 8 units
width = 6
Height = 5
To find the volume of a rectangular prism
[tex]V = a\times b \times c \: \: unit^3[/tex]
V = 8 x 6 x 5
V = 240 units^3.
Therefore, the volume of the prism is 240 units^3.
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Jeff's salary is 25% higher than Josh's. By how many percents is Josh's salary less than Jeff's?
PLEASE HELP HAVE TO GET DONE BY TODAY PLEASE!!!!!!!!!!!!!!!!!!
Answer:
Josh's salary is 20% less than Jeff's.
Step-by-step explanation:
Let
x----> Jeff's salary
y----> Josh's salary
we know that
x=1.25y
Solve for y
y=(1/1.25)x
y=0.8x
therefore
Josh's salary is 80% of Jeff's salary
or
Josh's salary is 20% less than Jeff's.
Nancy found that x = 1 is one solution to the quadratic equation (x + 2)2 = a. What is the value of a?
Answer:
9
Step-by-step explanation: (PLZ GIVE ME BRAINLIEST!!!! :))i did this question before
Answer:
The value of x is 9.
Step-by-step explanation:
Given equation,
[tex](x+2)^2=a[/tex]
If x = 1 is the solution of this equation,
Then it will satisfy the equation,
[tex]\implies (1+2)^2=a[/tex]
[tex]\implies (3)^2=a[/tex]
[tex]\implies a = 9[/tex]
Hence, the value of x is 9 if x = 1 is the solution of the given equation.
A square piece of plastic has sides that are 4 centimeters long. What is the plastics perimeter?
Perimeter of a square = side*side
4*4=16cm^2
The perimeter of a square piece of plastic with each side measuring 4 centimeters is 16 centimeters.
Explanation:The question is asking for the perimeter of a square piece of plastic with sides that are 4 centimeters long. To calculate the perimeter of a square, you multiply the length of one side by four. Since a square has four equal sides, the calculation in this case would be 4 cm (side length) times 4 (number of sides), which equals 16 cm. So, the perimeter of the square piece of plastic is 16 centimeters.
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Paula is going to choose the size, color, phrase, and picture for a birthday card for her friend. There are 2 sizes, 4 colors, 7 phrases, and 4 pictures for her to choose from. (The printing company charges a fee to add extra design elements, so she will choose only one of each.)
Answer:
Step-by-step explanation:
maybe 2 by 4 and 7 by 4 then add sorry if i'm wrong
Graph the system of equations. f(x)=−x^2+2x+4g(x)=−x+4 Which statements are true about the solutions to the system of equations? Select each correct answer.
An ordered pair that is the solution to the system of equations lies in Quadrant I .
An ordered pair that is the solution to the system of equations lies in Quadrant III .
An ordered pair that is the solution to the system of equations lies on the y-axis.
The x-coordinate of a solution to the system of equation is 3.
The y-coordinate of a solution to the system of equations is 4.
The y-coordinate of a solution to the system of equations is 0.
ANSWER
An ordered pair that is the solution to the system of equations lies on the y-axis.
The y-coordinate of a solution to the system of equations is 4.
EXPLANATION
The given system has equations:
[tex]y = {x}^{2} + 2x + 4[/tex]
[tex]y = - x + 4[/tex]
We equate both equations to get:
[tex] {x}^{2} + 2x + 4 = - x + 4[/tex]
This implies that,
[tex] {x}^{2} + 2x + x + 4 - 4 = 0[/tex]
[tex] {x}^{2} + 3x = 0[/tex]
[tex]x(x + 3) = 0[/tex]
[tex]x = 0 \: or \: x = - 3[/tex]
When x=0, y=-(0)+4=4
When x=-3, y=-(-3)+4=7
The solutions are: (0,4) and (-3,7)
The true statements about the system of equations are:
(a) An ordered pair that is the solution to the system of equations lies in Quadrant I .(c) An ordered pair that is the solution to the system of equations lies on the y-axis.(d) The x-coordinate of a solution to the system of equation is 3.(e) The y-coordinate of a solution to the system of equations is 4.The system of equations is given as:
f(x)=−x^2+2x+4
g(x)=−x+4
From the graph of the system of equations (see attachment), we have the following point of intersections
(x,y) = (0,4) and (3,1)
So, the true statements about the system of equations are:
(a), (c), (d) and (e)
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Suppose you roll a 6-faced die 90 times. About how many times would you expect to get a 5?
Answer: 15
Step-by-step explanation: because there are 6 sides. 1/6*90=15
Hoped this helped!
When rolling a fair six-sided die 90 times, the expected number of times you would roll a 5 is about 15, calculated by multiplying the probability of rolling a 5 (1/6) by the number of rolls (90).
If you roll a fair six-sided die 90 times, you would expect each number to appear with equal probability because the die is fair. For each individual roll, the probability of getting a 5 is 1 in 6, as there are six possible outcomes. To find the expected number of times you would get a 5 in 90 rolls, you simply multiply the probability of rolling a 5 by the total number of rolls.
Expected number of fives = Probability of rolling a 5 × Total number of rolls
= (1/6) × 90
= 15
Therefore, you would expect to roll a 5 approximately 15 times out of 90 rolls.
If AC is a diameter, which of the following must be a semicircle?
ABC
ACB
TCR
Answer:
abc is the one that must be a simicircle
Answer:
A. ABC.
Step-by-step explanation:
We have been given an image of a circle. We are asked to choose the arc, which must be a semicircle of our given circle.
Upon looking at our given diagram, we can see that AC divides our given circle into two semicircles ABC and ATC.
From our given choices, we can see that option A is the correct choice.
by car john traveled from his house to miami florida in 6 hours going 14 mph faster elliot can drive from his house to miami florida in 5 hours what is the distance from johns house to miami florida calculating spped distance and tiem
Answer:
420 miles
Step-by-step explanation:
Distance = rate × time
Let's say that the distance is D and that John's speed is S. We know that:
D = S × 6
When Elliot drives at speed S+14:
D = (S+14) × 5
Setting D = D:
S × 6 = (S+14) × 5
6S = 5S + 70
S = 70
Therefore, the distance D is:
D = 70 × 6
D = 420
The distance from John's house to Miami is 420 miles.
The distance from John's house to Miami, Florida is 420 miles, calculated by first determining John's speed using the given information and then applying the formula distance = speed times time.
To calculate the distance from John's house to Miami, Florida, we need to use the formula distance = speed times time. John's travel time is 6 hours. We are not directly given John's speed; instead, we know that Elliot drives 14 mph faster than John and it takes him 5 hours to cover the same distance. Let's denote John's speed as 's'. Therefore, Elliot's speed is 's + 14 mph'. Setting up an equation based on Elliot's travel information, we have distance = (s + 14 mph) times 5 hours.
Since they both travel the same distance, we can also write John's travel information as distance = s times 6 hours. Equating both expressions for distance, we get: s times 6 = (s + 14) times 5. Simplifying, we have 6s = 5s + 70, which leads to s = 70 mph. This means John's speed is 70 mph.
To find the distance to Miami, we return to the distance formula and substitute John's speed: distance = 70 mph times 6 hours = 420 miles. Hence, the distance from John's house to Miami, Florida is 420 miles.
A cylindrical shaped drum is used to store basketballs in a gymnasium. The hollow drum measures 48 inches high with a 24 inch radius. If the radius of a basketball is 6 inches, the maximum number of basketballs that the cylindrical drum contains is ______ (192, 48, 96)
Answer:
[tex]96\ basketballs[/tex]
Step-by-step explanation:
step 1
Find the volume of the cylinder (hollow drum)
The volume is equal to
[tex]V=\pi r^{2} h[/tex]
we have
[tex]h=48\ in[/tex]
[tex]r=24\ in[/tex]
substitute
[tex]V=\pi (24)^{2} (48)[/tex]
[tex]V=27,648\pi\ in^{3}[/tex]
step 2
Find the volume of one basketball
The volume of the sphere is equal to
[tex]V=\frac{4}{3}\pi r^{3}[/tex]
we have
[tex]r=6\ in[/tex]
substitute
[tex]V=\frac{4}{3}\pi (6)^{3}[/tex]
[tex]V=288\pi\ in^{3}[/tex]
step 3
Find the maximum number of basketballs that the cylindrical drum contains
so
Divide the volume of the cylinder by the volume of one basketball
[tex]27,648\pi/288\pi=96\ basketballs[/tex]
Point of tangency of an inscribed circle divides a leg of an isosceles triangle into 3 cm and 4 cm line segments (considered from the vertex to the base). Find the perimeter of the triangle.
Answer:
22
Step-by-step explanation:
The perimeter of the isosceles triangle with a point of tangency that divides one leg into 3 cm and 4 cm segments is 22 cm, having both legs 7 cm each and the base 8 cm.
Given an isosceles triangle with a point of tangency that divides one of the equal legs into segments of 3 cm and 4 cm, we can determine the perimeter of the triangle by first understanding the properties of such a triangle. The two legs are congruent, and by the Theorem 6, the radius from the center to the point of tangency is perpendicular to the tangent and bisects it. Knowing that the point of tangency divides a leg into two parts, we can denote the entire length of one leg as 3 cm + 4 cm, which equals 7 cm.
Since the triangle is isosceles, both legs are equal in length. This means the other leg is also 7 cm. To find the base, we recall the perpendicular from the center bisects it (Theorem 6). Hence, the base is twice one of the segments, either 3 cm or 4 cm. We will choose the longer segment to ensure that the vertex angles remain acute, and hence the base would be 2 * 4 cm = 8 cm.
Now the perimeter (P) of the triangle can be found by adding the lengths of the two legs and the base: P = 7 cm + 7 cm + 8 cm = 22 cm. Therefore, the perimeter of the isosceles triangle is 22 cm.