Answers
Answer:
(a) - [tex]y=-2x+18[/tex]
(b) - [tex]y=\frac{1}{2} x-\frac{9}{2}[/tex]
Step-by-step explanation:
Given the equation of a line, which we'll call line 1, find the following.
(a) - The equation of a line, which we'll call line 2, that is parallel to line 1 and travels through the point (9,0)
(b) - The equation of a line, which we'll call line 3, that is perpendicular to line 1 and travels through the point (9,0)
Given:
[tex]3y+6x=-6[/tex]
(1) - Write the equation of line 1 in slope-intercept form
[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Slope-Intercept Form:}}\\y=mx+b\\\bullet \ m \ \text{is the slope of the line}\\\bullet \ b \ \text{is the y-intercept of the line}\end{array}\right}[/tex]
[tex]3y+6x=-6\\\\\Longrightarrow 3y=-6-6x\\\\\Longrightarrow y=-\frac{6}{3} -\frac{6}{3}x \\\\\therefore \boxed{y=-2x-2}[/tex]
Thus, we can conclude the slope of line 1 is -2.
(2) - Answering part (a)
To find a line that is parallel to line 1, the slopes must be the same, -2. Use the point-slope form for a line to find the equation for line 2.
[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Point-Slope Form:}}\\y-y_1=m(x-x_1)\\\bullet \ m \ \text{is the slope of the line}\\\bullet \ (x_1,y_1) \ \text{is a point the line passes through}\end{array}\right}[/tex]
[tex]y-y_1=m(x-x_1); \ \text{Recall that} \ m=-2 \ \text{and} \ (x_1,y_1)=(9,0)\\\\\Longrightarrow y-0=-2(x-9)\\\\\therefore \boxed{\boxed{ y=-2x+18}}[/tex]
Thus, the equation for line 2 is found.
(2) - Answering part (b)
To find a line that is perpendicular to line 1, the slope of line 3 must be the opposite-reciprocal of line 1's. Once again, use the point-slope form of a line to find the equation of line 3.
[tex]y-y_1=m(x-x_1); \ \text{Recall that} \ m=\frac{1}{2} \ \text{and} \ (x_1,y_1)=(9,0)\\\\\Longrightarrow y-0=\frac{1}{2}(x-9)\\\\\therefore \boxed{\boxed{ y=\frac{1}{2} x-\frac{9}{2} }}[/tex]
Thus, the equation of line 3 is found.
Related Questions
What change in volume results if 60.0 mL of gas is cooled from 33.0°C to 5.00°C
Answers
Answer:
The change in volume is -5.5 mL (a decrease in volume of 5.5 mL) when 60.0 mL of gas is cooled from 33.0°C to 5.00°C.
Step-by-step explanation:
To calculate the change in volume, we need to use the ideal gas law equation:
V1/T1 = V2/T2
where V1 and T1 are the initial volume and temperature, and V2 and T2 are the final volume and temperature.
Given:
V1 = 60.0 mL
T1 = 33.0°C = 33.0 + 273.15 = 306.15 K
T2 = 5.00°C = 5.00 + 273.15 = 278.15 K
Now we can calculate V2, the final volume:
V1/T1 = V2/T2
(60.0 mL) / (306.15 K) = V2 / (278.15 K)
Cross-multiplying and solving for V2:
V2 = (60.0 mL) * (278.15 K) / (306.15 K)
V2 = 54.5 mL
The final volume, V2, is 54.5 mL.
To find the change in volume, we subtract the initial volume from the final volume:
Change in volume = V2 - V1
Change in volume = 54.5 mL - 60.0 mL
Change in volume = -5.5 mL
Therefore, the change in volume is -5.5 mL (a decrease in volume of 5.5 mL) when 60.0 mL of gas is cooled from 33.0°C to 5.00°C.
Find the area of each regular polygon. Round your
answer to the nearest tenth if necessary.
10.4
661.5
364
O 462.7
533.6
10
Answers
Area of the given heptagon = 364 square unit.
The given figure is regular heptagon,
length of each side = 10
Distance between Centre and edge = 10.4
Since we know that,
A seven-sided polygon with seven angles, seven vertices, and seven edges is known as a heptagon. They may have the same or different length dimensions. It is a closed figure, and a regular heptagon is one with all equal seven sides.
Now,
Perimeter of the given heptagon = 7x10
= 70
Area of heptagon = perimeter x 10.4/2
= 70 x 5.2
= 364
Hence,
Area = 364 square unit.
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a trucker is driving on the highway traveling 65 miles per hour write an equation that shows the relationship between the number of hours of highway travel x and the number of miles traveled 7 answer quickly for 20 points!!
Answers
y = 65x
In a class of students, the following data table summarizes how many students play
an instrument or a sport. What is the probability that a student chosen randomly
from the class plays a sport and an instrument?
Plays a sport
Does not play a sport
Plays an instrument Does not play an instrument
10
3
8
2
Answers
50% probability that a student chosen randomly from the class does not play a sport.
Using the probability concept, it is found that there is a 0.5 = 50% probability that a student chosen randomly from the class does not play a sport.
A probability is the number of desired outcomes divided by the number of total outcomes.
In this problem:
In total, there are 8 + 7 + 3 + 12 = 30 students.
Of this total, 3 + 12 = 15 do not play a sport.
Thus, probability = 15/30
= 1/2
= 0.5
Therefore, 50% probability that a student chosen randomly from the class does not play a sport.
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Compare the functions shown below:
f(x)
polynomial graph with x intercepts at negative 3, negative 2, 1, 2, y intercept at negative 12
g(x)
trig graph with points at 0, 3 and pi over 2, 0 and pi, negative 3 and 3 pi over 2, 0 and 2 pi, 3
Over the interval x = 0 to x = 2π
h(x) = 2x3 − 2x2 − 3x + 2
Which function has the most x-intercepts? (2 points)
f(x)
g(x)
h(x)
All three functions have the same number of x-intercepts.
Answers
f(x)
Answer:
f(x) has the most x intercepts
write an explicit formula for an the nth term of the sequence 6, 12, 18
Answers
Answer:
[tex]a_{n}[/tex] = 6n
Step-by-step explanation:
there is a common difference between consecutive terms, that is
12 - 6 = 18 - 12 = 6
this indicates the sequence is arithmetic with nth term
[tex]a_{n}[/tex] = a₁ + d(n - 1)
where a₁ is the first term and d the common difference
here a₁ = 6 and d = 6 , then
[tex]a_{n}[/tex] = 6 + 6(n - 1) = 6 + 6n - 6 = 6n
Determine the number of solutions:
10-
8-
O No Solution
6
4
2-
++++
-10-8-6-4-2,
N
पं
-6
-8
-10-
O Infinitely Many Solutions
One Solution
6x-2y=-4
3x-y=-2
2
8 10
Answers
Answer:
infinitely many solutions
Step-by-step explanation:
let's compare each equations to equation of a line
which is y=mx+c
first equation
6x - 2y = -4
-2y = -6x -4
divide through by -2
y=3x+2
second equation
3x-y= -2
-y= -3x-2
divide through by -1
y= 3x+2
since the slope and intercept of the first and second equation equal each other then the number of solutions is infinitely many.
note that m is slope and c is intercept.
50 Points! Multiple choice geometry question. Photo attached. Thank you!
Answers
The value of x is 8 units.
Given that a right triangle, with an altitude dividing the hypotenuse into 4 and 16 units,
The measure of the altitude is x we need to find the measure of the x,
So, according to the definition of similarity,
Their corresponding sides are proportional,
We get,
x / 4 = 16 / x
x² = 4 × 16
x² = 64
x = 8
Hence the value of x is 8 units.
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A sample of 311 people is selected. The people are classified according to place of residence ("urban", "suburban", or "rural"). They are also classified according to highest educational degree earned ("no college degree", "two-year degree", "four-year degree", or "advanced degree"). The results are given in the contingency table below. Urban Suburban Rural No college degree Two-year degree Four-year degree 34 42 22 35 21 22 16 34 16 Advanced degree 23 23 23 What is the relative frequency of people in the sample whose place of residence is suburban and whose highest degree is a two-year degree? Round your answer to two decimal places.
Answers
Answer: 0.07
Step-by-step explanation:
To find the relative frequency of people in the sample whose place of residence is suburban and whose highest degree is a two-year degree, we need to calculate the ratio of the number of people with those characteristics to the total sample size.
Looking at the contingency table, we can see that the number of people in the suburban category with a two-year degree is 21.
The total sample size is given as 311.
Therefore, the relative frequency can be calculated as:
Relative frequency = (Number of people in suburban category with two-year degree) / (Total sample size)
= 21 / 311
≈ 0.0676
Rounded to two decimal places, the relative frequency is approximately 0.07.
Simplify 6 5/8 + 6 5/6
Answers
The required sum of mixed fraction answer is 12 35/24.
Given that 6 5/8 + 6 5/6.
To find the sum of two or more mixed fractions, add the whole part of two fractions, add the fraction part of two fractions, then add these sums.
The sum (s1) of whole parts of two fractions is
s1= 6 + 6 = 12.
The sum(s2) of fraction parts of two fractions is
s2 = 5/8 +5/6.
L.C.M of 8 and 6 is 24.
Multiply each fraction by 12/12 gives,
s2 = 5/8 × 24/24 + 5/6 × 24/24.
s2 = 15/24 + 20/24.
s2 = 35/24.
The sum of mixed fractions is s1 + s2
Sum = 12 + 35/24
Thus, the required sum of mixed fraction is 12 35/24.
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An 80-ounce bottle of apple juice contains 32 ounces of water.
Answers
Using the proportion principle , Orange Juice has a greater percentage of water than Apple juice
Calculating ProportionsTo determine which juice has a greater percent of water, we need to calculate the percentage of water in each juice.
Apple juicePercentage of water = (Water content / Total content) * 100
Water content = 32 ounces
Total content = 80 ounces
Percentage of water in apple juice = (32 / 80) * 100 = 40%
Orange juiceWater content = 48 ounces
Total content = 64 ounces
Percentage of water in orange juice = (48 / 64) * 100 = 75%
Comparing the percentages, we can conclude that orange juice has a greater percentage of water (75%) than the apple juice (40%).
Therefore, the orange juice contains a higher percentage of water.
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Complete question :
an 80 ounce bottle of apple juice contains 32 ounces of water a 64 ounce bottle of orange juice has 48 ounces of water which juice has a greater percent of water
Which polynomial function could be represented by the graph below?
Answers
Circle the error in the problem and rewrite what the correct step should be
Answers
The error in the problem is given by the equation in step(3). and the value is given by g ( x ) = 2x² + 4x - 4
Given data ,
Let the equation be represented as g ( x )
Now , the value of g ( x ) is
g ( x ) = 2 ( x + 1 )² - 6
On simplifying the equation , we get
From the distributive law of multiplication , we get
g ( x ) = 2 ( x + 1 ) ( x + 1 ) - 6
g ( x ) = 2 ( x² + 2x + 1 ) - 6
On further simplification , we get
g ( x ) = 2x² + 4x + 2 - 6
g ( x ) = 2x² + 4x - 4
Therefore, the value of g ( x ) is g ( x ) = 2x² + 4x - 4
Hence , the equation is g ( x ) = 2x² + 4x - 4
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Pizza General charges 10 dollars per medium pizza and a 5-dollar delivery fee. Fill in the table to show how much you would pay to get each number of pizzas delivered. Make sure to consider both the cost of the pizzas and the delivery fee.NOWWWWW
Answers
Answer:
Step-by-step explanation:
$15 for one pizza it's simple
the value stock in 1940 is $1.25. it’s value grows by 7% each year after 1940. write an equation representing the value of the stock V(t), in dollars, t years after 1940
Answers
This equation represents the value of the stock, V(t), in dollars, t years after 1940, taking into account the 7% annual Growth rate V(t) = $1.25 * (1 + 0.07)^t.
The value of the stock, V(t), in dollars, t years after 1940, we can use an exponential growth equation since the stock's value grows by 7% each year after 1940.
formulate the equation step by step:
1. We know that the initial value of the stock in 1940 is $1.25. This initial value serves as the starting point for our equation.
2.We are given that the value of the stock grows by 7% each year. This means that for every year that passes, the stock's value increases by 7% of its previous value.
3. In mathematical terms, if V(0) represents the initial value in 1940 ($1.25), then the value of the stock after t years can be represented as:
V(t) = V(0) * (1 + r)^t,
where r is the growth rate expressed as a decimal (in this case, 7% = 0.07).
4. Plugging in the given values, our equation becomes:
V(t) = $1.25 * (1 + 0.07)^t.
This equation represents the value of the stock, V(t), in dollars, t years after 1940, taking into account the 7% annual growth rate.
For example, if we wanted to find the value of the stock 10 years after 1940, we would substitute t = 10 into the equation:
V(10) = $1.25 * (1 + 0.07)^10.
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PLEASE ANSWER ASAP!!!!!
Answers
Answer:
[tex]\huge\boxed{\sf r = 5}[/tex]
Step-by-step explanation:
Given that,
7(q + 5) = (q + r)7Distribute7q + 35 = 7q + 7r
Subtract 7q from both sides7q - 7q + 35 = 7q - 7q + 7r
35 = 7r
Divide both sides by 735/7 = r
5 = r
OR
r = 5[tex]\rule[225]{225}{2}[/tex]
Answer:
r = 5
Step-by-step explanation:
Given statement,
→ 7(q + 5) is equivalent to (q + r)7.
Forming the equation,
→ 7(q + 5) = 7(q + r)
Now we have to,
→ Find the required value of r.
Then the value of r will be,
→ 7(q + 5) = 7(q + r)
Applying Distributive property:
→ 7(q) + 7(5) = 7(q) + 7(r)
→ 7q + 35 = 7q + 7r
Cancelling 7q from both sides:
→ 35 = 7r
→ 7r = 35
Dividing the RHS with number 7:
→ r = 35/7
→ [ r = 5 ]
Therefore, the value of r is 5.
The perimeter of an isosceles triangle is 45 inches. Two sides of the triangle are congruent,
and the third side is twice as long as the sum of the congruent sides. What is the length of the
third side of the triangle in inches?
Answers
Answer:
Length of third side = 30 inches
Step-by-step explanation:
We will need a system of equations to find the length of the third side.
First equation:
Let x represent the length of just one of the congruent sides and let y represent the length of the third side.We know the perimeter of a triangle is simply the sum of its sides.
Thus, since the perimeter is 45, our first equation is x + x + y = 45, which simplifies to 2x + y = 45.
Second equation:
Since we're told that the length of the third side is twice as long as the sum of the congruent sides, our second equation is y = 2(x + x), which simplifies to y = 2(2x) and then finally y = 4x
Method: Substitution.
We can first solve for x (length of one of the congruent sides) by substituting y = 4x for y in 2x + y = 45:
2x + 4x = 45
6x = 45
x = 7.5
Find y:
Now we can find y (length of the third side) by plugging in 7.5 for x in y = 4x:
y = 4(7.5)
y = 30
Thus, the length of the third side is 30 inches.
Optional Step to check validity of answers.
First equation: Check that the sum of 7.5, 7.5, and 30 = 45
7.5 + 7.5 + 30 = 45
15 + 30 = 45
45 = 45
Second equation: Check that 30 is twice the sum of 7.5 + 7.5
30 = 2(7.5 + 7.5)
30 = 2(15)
30 = 30
Thus, our answers are correct and we've correctly determined the length of the third side of the triangle in inches.
50 Points! Multiple choice geometry question. Photo attached. Thank you!
Answers
The calculated measure of the angle 1 is 43 degrees
How to determine the measure of the angle 1From the question, we have the following parameters that can be used in our computation:
The circle
Also, we have
The line AB tangent to the circle
This means that
Angle 1 = 1/2 * the arc subtended by the line
So, we have
Angle 1 = 1/2 * 86
Evaluate
Angle 1 = 43
Hence, the measure of the angle 1 is 43 degrees
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6.56*10^-7 by 4.86*10^-7
Answers
6.56 × [tex]10^{-7}[/tex] by 4.86×[tex]10^{-7}[/tex] is 3.18816 × [tex]10^{-13}[/tex] is factorize.
To find the product of 6.56 × [tex]10^{-7}[/tex] by 4.86×[tex]10^{-7}[/tex]. The conventions of scientific notation can be applied. To get 31.8816, we must first multiply the coefficients, which are 6.56 and 4.86. The exponents, which are -7 and -7, are then added to provide -14.
The product of 6.56 × [tex]10^{-7}[/tex] by 4.86 × [tex]10^{-7}[/tex] is 3.18816 × [tex]10^{-13}[/tex] This, according to the conventions of scientific notation, can be calculated by multiplying the coefficients and adding the exponents.
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NEED HELP ASAP WILL GIVE BRAINLIEST HELP!
Answers
If Jeremy wants to try to prove ∠3 ≅ ∠6, then the statement that Jeremy can write is ∠1 ≅ ∠4
That's it :)
Over a period of one year, a retailer sells widgets at 11
different prices. He Calculates the average number of oiunds sold per day at each different price. From theese data, the following are calculated.
Answers
The regression equation is: y = -0.0068824x + 36.8881
So, the correct option is:
D) None of these.
To compute the regression equation based on the given data, we can use the formulas for slope (b1) and y-intercept (bo):
b1 = (NΣXY - ΣXΣY) / [tex](N\sum X^2 - (\sum X)^2)[/tex]
bo = (ΣY - b1ΣX) / N
Where:
N = Number of data points (in this case, 11)
ΣX = Sum of all X values (prices)
ΣY = Sum of all Y values (average number of pounds sold per day)
ΣXY = Sum of the product of X and Y values
[tex]\sum X^2[/tex] = Sum of the squares of X values
Using the provided data:
Ex = 210
Xy = 450
Zcy = 10387
Zz = 7275
Xy = 42471
Let's calculate the required values:
N = 11
ΣX = Ex + Zcy + Zz = 210 + 10387 + 7275 = 17872
ΣY = Xy = 450
ΣXY = Xy = 450
[tex]\sum X^2 = (Ex)^2 + (Zcy)^2 + (Zz)^2 = (210)^2 + (10387)^2 + (7275)^2 = 239208144[/tex]
Now, substitute these values into the formulas to find b1 and bo:
[tex]b1 = (11 \times450 - 17872 \times 450) / (11 \times239208144 - (17872)^2)[/tex]
= -14112250 / 2050445128
≈ -0.0068824
bo = (450 - (-0.0068824 [tex]\times[/tex] 17872)) / 11
= (450 + 122.7790368) / 11
≈ 36.8881
Therefore, the regression equation is:
y = -0.0068824x + 36.8881
So, the correct option is:
D) None of these.
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The complete question may be like:
Over a period of one year, retailer sells widgets at 11 different prices. He Calculates the average number of oiunds sold per day at each different price. From theese data, the following are calculated. Ex = 210, Xy = 450, Zcy = 10387, Zz? 7275 Xy? 42471 Compute the regression equation , SELEC A) y 0.20261x + 0.549951 B) y = 0.549951x + 30.410021 C) y = 0.20261. + 30.410021 D) none of these =b1x + bo.
The difference between an observational study and an experiment is thatin an observational study, only one group is studied, and in an experiment, two groups are studied.in an observational study, the researchers do not control treatment, and in an experiment, they do.in an experiment, cause-and-effect is analyzed, and in an observational study, it is not.in an experiment, one group is studied over a short period of time, and in an observational study, the group is studied over a longer period of time.
Answers
The difference between an observational study and an experiment is that in an observational study, the researchers do not control treatment, and in an experiment, they do
What is observational study and an experiment?In an observational study, it should be noted that the participants are measured or surveyed without any attempt to influence them. *
However the controlled experiment, participants or objects are divided into groups, and one group is given a treatment while the other is not.
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Enter the number that belongs in the green box
Answers
The number in the green box is 31.78° .
Given,
Triangle with sides 11 , 12 , 20.
From Cosine Law.
Let △ABC have sides a, b, c.
Here,
a = 11
b = 20
c = 12
Length of the side opposite vertex A is called a.
Length of the side opposite vertex B is called c.
Length of the side opposite vertex C is called c.
Let θ = ∠C
Then, the Cosine Law says that
c²=a²+b²−2abcosθ.
Substitute the values of a , b , c .
12² = 11² + 20² - 2 × 11× 20 cosθ
θ = 31.78°
Hence the angle can be found out from cosine law.
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In the figure below, j Il m. Find the values of y and z.
78°
11
>
(4z - 14)
y=
Z =
Answers
The values of the variables are y = 102° and z = 23°.
Given that are two parallel lines j and m we need to find the measures of the variables y and z,
So,
Since, j║m, therefore, y and 78° are supplementary because they are exterior consecutive angles between two parallel lines,
So,
y + 78° = 180°
y = 180° - 78°
y = 102°
Now,
y and (4z-14) are linear pair angles, which means they are also supplementary.
Therefore,
y + (4z-14) = 180°
102° + 4z - 14 = 180°
4z + 88 = 180°
4z = 92°
z = 23°
Hence the values of the variables are y = 102° and z = 23°.
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50 Points! Multiple choice geometry question. Photo attached. Thank you!
Answers
The area of the slice of pizza is 25.1 inches squared.
How to find the area of the slice of pizza?The circular pizza has a diameter of 16 inches. Each slice of pizza have a angle of 45 degrees.
Therefore, the area of the slice of pizza can be calculated as follows:
area of the slice of pizza = ∅ / 360 × πr²
where
r = radius∅ = central angleTherefore,
r = 16 /2
r = 8 inches
area of the slice of pizza = 45 / 360 × 3.14 × 8²
area of the slice of pizza = 45 / 360 × 3.14 × 64
area of the slice of pizza = 9043.2 / 360
area of the slice of pizza = 25.12 inches²
area of the slice of pizza = 25.1 inches²
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how many cubic inches will a rectangular pyramid hold if its 15 in height by 12 inches base
Answers
The number of cubic inches that the rectangular pyramid will hold if it has 15 inches height and 12 inches base is 720 inches³.
Here we have to find the volume of the pyramid since cubic unit means the volume.
We know that,
Volume of any pyramid = 1/3 × Base area × Height
Here the given pyramid is a rectangular pyramid.
Base of the pyramid will be a rectangle.
So, base area = length × width
Now given that,
Base is 12 inches.
Since there is only a dimension, width and length will be equal to 12 inches.
So base area = 12 × 12 = 144 inches²
Height = 15 inches
Volume of the pyramid = 1/3 × 144 × 15
= 720 inches³
Hence the volume is 720 inches³.
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Which expression can be used to find the value of x?
(sin 29°) (sin 42°)
9
O
○ 9(sin 29°) (sin 42°)
O
9(sin 29°)
sin 42°
9(sin 42°)
sin 29°
Answers
Answer:
Step-by-step explanation:
what is the value of z (2x+6) degrees 12 degrees
Answers
The value of x indicated in the right angle of EFG is determined as 36.
option D.
What is the value of x?The value of x is calculated by applying the principle of sum of angles in a right angle triangle as follows;
A right triangle has a right angle and two angles that are complementary.
If angle EFG is a right angle, then the sum of angles at point F is 90 degrees.
The value of x is calculated by setting up the following equations;
2x + 6 + 12 = 90
2x + 18 = 90
2x = 90 - 18
2x = 72
x = 72/2
x = 36
Thus, the value of x is calculated by applying the principle of sum of angles in a right angle triangle.
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Which expressions are less than 7/8? Choose2 answers 7/8 x 9/19 7/8 x 3/4 7/8 x 9/9 7/8 x 4/3
Answers
The two expressions that are less than 7/8 are 7/8 x 9/19 and 7/8 x 3/4.
To determine which expressions are less than 7/8, we can compare them individually to 7/8 and evaluate their values.
Let's consider each expression one by one:
7/8 x 9/19:
To compare this expression to 7/8, we multiply the fractions:
(7/8) x (9/19) = 63/152.
Since 63/152 is less than 7/8, this expression is less than 7/8.
7/8 x 3/4:
Multiplying the fractions:
(7/8) x (3/4) = 21/32.
Since 21/32 is less than 7/8, this expression is less than 7/8.
7/8 x 9/9:
The fraction 9/9 simplifies to 1, so the expression becomes:
(7/8) x 1 = 7/8.
Since 7/8 is equal to 7/8 (not less than), this expression is not less than 7/8.
7/8 x 4/3:
Multiplying the fractions:
(7/8) x (4/3) = 28/24.
We can simplify 28/24 to 7/6.
Since 7/6 is greater than 7/8, this expression is not less than 7/8.
Therefore, the two expressions that are less than 7/8 are 7/8 x 9/19 and 7/8 x 3/4.
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Lucas wants to create a triangle with sides measuring 9 inches 13 inches and 15 inches. He says that more than one triangle is possible given the side lengths. What time about Lucas claim is true?
Answers
The Triangle Inequality Theorem holds for these values, and only one triangle is possible with these side lengths. Lucas' claim is false.
What is Triangle Inequality Theorem ?According to the Triangle Inequality Theorem, any two triangle sides' sums must be greater than the length of the third side. In other words, the following three inequalities must hold for any three positive numbers, a, b, and c:
a+b>c
a+c>b
b+c>a
In this case, a=9, b=13, and c=15. Let's check if the Triangle Inequality Theorem holds for these values:
9+13>15 (true)
9+15>13 (true)
13+15>9 (true)
Therefore, the Triangle Inequality Theorem holds for these values, and only one triangle is possible with these side lengths.
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