Similar Triangles
Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to
prove the triangles similar? Explain your reasoning.
I need help on number 1 and 2
Answers
The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;
1. ΔAJK ~ ΔSWY according to the SAS similarity postulate
2. ΔLMN ~ ΔLPQ according to the AA similarity postulate
3. ΔPQN ~ ΔLMN
LM = 12, QP = 8
4. ΔLMK~ΔLNJ
NL = 21, ML = 14
What are similar triangles?
Similar triangles are triangles that have the same shape but may have different sizes.
1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;
24/16 = 3/2
18/12 = 3/2
The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.
2. The ratio of the corresponding sides in each of the triangles are;
MN/LN = 8/10 = 4/5
PQ/LQ = 12/(10 + 5) = 12/15 = 4/5
The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule
3. The alternate interior angles theorem indicates;
Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;
ΔPQN ~ ΔLMN by the AA similarity postulate
LM/QP = (x + 3)/(x - 1) = 18/12
12·x + 36 = 18·x - 18
18·x - 12·x = 36 + 18 = 54
6·x = 54
x = 54/6 = 9
LM = 9 + 3 = 12
QP = x - 1
QP = 9 - 1 = 8
4. The similar triangles are; ΔLMK and ΔLNJ
ΔLMK ~ ΔLNJ by AA similarity postulate
ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)
ML/NL = LK/LJ = (24 - 8)/24
(24 - 8)/24 = (6·x + 2)/((7·x + 7)
16/24 = (6·x + 2)/(7·x + 7)
16 × (7·x + 7) = 24 × (6·x + 2)
112·x + 112 = 144·x + 48
144·x - 112·x = 32·x = 112 - 48 = 64
x = 64/32 = 2
ML = 6 × 2 + 2 = 14
NL = 7 × 2 + 7 = 21
MN = 2 + 5 = 7
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Related Questions
Graph the function f(x)= 3+2 in x and its inverse from model 1.
Answers
The graph of the function and its inverse is added as an attachment
Sketching the graph of the function and its inverseFrom the question, we have the following parameters that can be used in our computation:
f(x) = 3 + 2ln(x)
Express as an equation
So, we have
y = 3 + 2ln(x)
Swap x and y in the above equation
x = 3 + 2ln(y)
Next, we have
2ln(y) = x - 3
Divide by 2
ln(y) = (x - 3)/2
Take the exponent of both sides
[tex]y = e^{\frac{x - 3}{2}}[/tex]
Next, we plot the graphs
The graph of the functions is added as an attachment
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how can you write the expression with a rationalized denominator?
3 sqrt 2 / 3 sqrt 6
see photo attached for answers
Answers
The expression (3√2) / (3√6) with a rationalized denominator is 3√9 / 6. Option C is the correct answer.
To rationalize the denominator in the expression (3√2) / (3√6), we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of √6 is -√6, so we multiply the expression by (-√6) / (-√6):
(3√2 / 3√6) * (-√6 / -√6)
This simplifies to:
-3√12 / (-3√36)
Further simplifying, we have:
-3√12 / (-3 * 6)
-3√12 / -18
Finally, we can cancel out the common factor of 3:
- 3√9 / - 6.
Simplifying further, we get:
3√9 / 6.
Option C is the correct answer.
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What is the z score for Brazil?
Answers
The z-score for Brazil is given as follows:
Z = 0.87.
What is the z-score formula?The z-score formula is defined as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
In which:
X is the measure.[tex]\mu[/tex] is the population mean.[tex]\sigma[/tex] is the population standard deviation.The parameters for this problem are given as follows:
[tex]X = 6.24, \mu = 4.8, \sigma = 1.66[/tex]
Hence the z-score for Brazil is given as follows:
Z = (6.24 - 4.8)/1.66
Z = 0.87.
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A village P is 12 km from village Q. It takes 3 hours 20 minutes to travel from Q to P and back to Q by a boat. If the boat travels at a speed of 6 km/h from P to Q and (6 + x) km/h back to P, find the value of x.
Answers
Answer:
Hope this helps and have a nice day
Step-by-step explanation:
To find the value of x, we can use the formula:
Time = Distance / Speed
Let's calculate the time taken to travel from Q to P and back to Q.
From Q to P:
Distance = 12 km
Speed = 6 km/h
Time taken from Q to P = Distance / Speed = 12 km / 6 km/h = 2 hours
From P to Q:
Distance = 12 km
Speed = (6 + x) km/h
Time taken from P to Q = Distance / Speed = 12 km / (6 + x) km/h
Given that the total time taken for the round trip is 3 hours 20 minutes, we can convert it to hours:
Total time = 3 hours + (20 minutes / 60) hours = 3 + (1/3) hours = 10/3 hours
According to the problem, the total time is the sum of the time from Q to P and from P to Q:
Total time = Time taken from Q to P + Time taken from P to Q
Substituting the values:
10/3 hours = 2 hours + 12 km / (6 + x) km/h
Simplifying the equation:
10/3 = 2 + 12 / (6 + x)
Multiply both sides by (6 + x) to eliminate the denominator:
10(6 + x) = 2(6 + x) + 12
60 + 10x = 12 + 2x + 12
Collecting like terms:
8x = 24
Dividing both sides by 8:
x = 3
Therefore, the value of x is 3.
Answer:
x = 3
Step-by-step explanation:
speed = distance / time
time = distance / speed
Total time from P to Q to P:
T = 3h 20min
P to Q :
s = 6 km/h
d = 12 km
t = d/s
= 12/6
t = 2 h
time remaining t₁ = T - t
= 3h 20min - 2h
= 1 hr 20 min
= 60 + 20 min
= 80 min
t₁ = 80/60 hr
Q to P:
d₁ = 12km
t₁ = 80/60 hr
s₁ = d/t₁
[tex]= \frac{12}{\frac{80}{60} }\\ \\= \frac{12*60}{80}[/tex]
= 9
s₁ = 9 km/h
From question, s₁ = (6 + x)km/h
⇒ 6 + x = 9
⇒ x = 3
What are the dimensions of the rectangle shown on the coordinate plane?
The base is 5 units and the height is 3 units.
The base is 4 units and the height is 7 units.
The base is 7 units and the height is 5 units.
The base is 7 units and the height is 3 units.
Answers
Answer:
D The base is 7 units and the height is 3 units.
Step-by-step explanation:
The answer is d I counted the width/base then the height/length and found answer.
A company Charting its profits notices that the relationship between the number of units sold,x, and the profit,P, is a linear. If 170 units sold results in $20 profit and 220 units sold results in $2820 profit, write the profit function for this company.
P=
Find the marginal profit
$
Answers
Step-by-step explanation:
a linear relationship or function is described in general as
y = f(x) = ax + b
Because the variable term has the variable x only with the exponent 1, this makes this a straight line - hence the name "linear".
here f(x) is P(x) :
P(x) = ax + b
now we are using both given points (ordered pairs) to calculate a and b :
20 = a×170 + b
2820 = a×220 + b
to eliminate first one variable we subtract equation 1 from equation 2 :
2800 = a×50
a = 2800/50 = 280/5 = 56
now, we use that in any of the 2 original equations to get b :
20 = 56×170 + b
b = 20 - 56×170 = 20 - 9520 = -9500
so,
P(x) = 56x - 9500
Team A and Team B together won 50% more games than Team C did. Team A won 50% as many games as Team B did. The three teams won 60 games in all. How many games did each team win?
Answers
Let x be the number of games won by Team A.
Let y be the number of games won by Team B.
Let z be the number of games won by Team C.
From the given information, we can form the following equations:
Equation 1: x + y + z = 60 (The total number of games won by the three teams is 60.)
Equation 2: x = (1/2)y (Team A won 50% as many games as Team B.)
Equation 3: x + y = 1.5z (Team A and Team B together won 50% more games than Team C.)
Now, let's solve this system of equations:
Substituting Equation 2 into Equation 3, we get:
(1/2)y + y = 1.5z
(3/2)y = 1.5z
y = (1.5z) * (2/3)
y = z
Substituting y = z into Equation 1, we have:
x + y + z = 60
x + y + y = 60
x + 2y = 60
Substituting y = z into Equation 3, we have:
x + y = 1.5z
x + y = 1.5y
x = 0.5y
Now, we can substitute x = 0.5y and y = z into Equation 1:
0.5y + 2y = 60
2.5y = 60
y = 60 / 2.5
y = 24
Substituting y = 24 into x = 0.5y:
x = 0.5 * 24
x = 12
Substituting y = 24 into the equation y = z:
z = 24
Therefore, Team A won 12 games, Team B won 24 games, and Team C won 24 games as well.
Which type of conic section is defined by the equation:... 100pts
Answers
Answer:
This is an equation of a parabola.
[tex](y+6)^2=4(x+1)[/tex]
Step-by-step explanation:
A conic section is a curve obtained by the intersection of a plane and a cone. The three major conic sections are parabola, hyperbola and ellipse (the circle is a special type of ellipse).
The standard equations for hyperbolas and ellipses all include x² and y² terms. The standard equation for a parabola includes the square of only one of the two variables.
Therefore, the equation y² - 4x + 12y + 32 = 0 represents a parabola, as there is no x² term.
As the y-variable is squared, the parabola is horizontal (sideways), and has an axis of symmetry parallel to the x-axis.
The conic form of a sideways parabola is:
[tex]\boxed{(y-k)^2=4p(x-h)}[/tex]
where:
(h, k) is the vertex.(h+p, k) is the focus.x = h-p is the directrix.To write the given equation in conic form, we need to complete the square for the y-variable.
Rearrange the equation so that the y-terms are on the left side:
[tex]y^2 + 12y = 4x - 32[/tex]
Add the square of half the coefficient of the y-term to both sides of the equation:
[tex]y^2 + 12y+\left(\dfrac{-12}{2}\right)^2 = 4x - 32+\left(\dfrac{-12}{2}\right)^2[/tex]
[tex]y^2 + 12y+\left(-6\right)^2 = 4x - 32+\left(-6\right)^2[/tex]
[tex]y^2 + 12y+36 = 4x - 32+36[/tex]
[tex]y^2 + 12y+36 = 4x +4[/tex]
Factor the perfect square trinomial on the left side of the equation:
[tex](y+6)^2=4x+4[/tex]
Factor out the coefficient of the x-term from the right side of the equation:
[tex](y+6)^2=4(x+1)[/tex]
Therefore, the equation of the given conic section in conic form is:
[tex]\boxed{(y+6)^2=4(x+1)}[/tex]
where:
(-1, -6) is the vertex.(0, -6) is the focus.x = -2 is the directrix.The conic section of the equation y² - 9x + 12y + 32 = 0 is a parabola
Selecting the conic section of the equationThe given equation is
y² - 9x + 12y + 32 = 0
The above equation is an illustration of a parabola equation
The standard form of a parabola is
(x - h)² = 4a(y - k)²
Where
(h, k) is the center
While the general form of the equation is
Ax² + Dx + Ey + F = 0
In this case, the equation y² - 9x + 12y + 32 = 0 takes the general form
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