In ∆ABC shown below, ∡BAC is congruent to ∡BCA. Given: Base ∡BAC and ∡ACB are congruent.
Prove: ∆ABC is an isosceles triangle.
Construct a perpendicular bisector from point B to line segment AC.
Label the point of intersection between this perpendicular bisector and line segment AC as point D.
m∡BDA and m∡BDC is 90° by the definition of a perpendicular bisector.
∡BDA is congruent to ∡BDC by the definition of congruent angles. Line segment AD is congruent to line segment DC by _______1________.
∆BAD is congruent to ∆BCD by the _______2________. Line segment AB is congruent to line segment BC because corresponding parts of congruent triangles are congruent (CPCTC).
Consequently, ∆ABC is isosceles by definition of an isosceles triangle.
options are:
a)1. Angle-Side-Angle (ASA) Postulate
2. corresponding parts of congruent triangles are congruent (CPCTC)
b) 1. corresponding parts of congruent triangles are congruent (CPCTC)
2. Angle-Side-Angle (ASA) Postulate
c) 1. the definition of a perpendicular bisector
2. Angle-Side-Angle (ASA) Postulate
d) 1. corresponding parts of congruent triangles are congruent (CPCTC)
2. the definition of a perpendicular bisector
Answers
Refer to the image attached.
Given: [tex]\angle BAC[/tex] and [tex]\angle BCA[/tex] are congruent.
To Prove: [tex]\Delta ABC[/tex] is an isosceles triangle.
Construction: Construct a perpendicular bisector from point B to line segment AC. Label the point of intersection between this perpendicular bisector and line segment AC as point D.
Proof:
Consider [tex]\Delta BDA, \Delta BDC[/tex]
[tex]\angle BDA = \angle BDC= 90^\circ[/tex]
(By the definition of perpendicular bisector)
[tex]AD=DC[/tex] (By the definition of perpendicular bisector)
So, Line segment AD is congruent to DC by the definition of perpendicular bisector.
[tex]\angle BAC[/tex] = [tex]\angle BCA[/tex] (given)
So, [tex]\Delta BDA \cong \Delta BDC[/tex] by ASA congruence postulate.
∆BAD is congruent to ∆BCD by the ASA congruence Postulate.
Line segment AB is congruent to line segment BC because corresponding parts of congruent triangles are congruent (CPCTC).
So, Option C is the correct answer.
Related Questions
You know that a certain trait occurs in 30% of the population. You want to know how many individuals you have to draw from the population, on average, to get three subjects with the trait.
Answers
To find out how many individuals you have to draw from the population, on average, to get three subjects with the trait, you can use the geometric distribution. The average number of trials until the third success can be calculated by dividing 1 by the probability of success.
Explanation:To find out how many individuals you have to draw from the population, on average, to get three subjects with the trait, you can use the geometric distribution.
The probability of success (finding an individual with the trait) is 30% or 0.30, and the probability of failure is 70% or 0.70.
To calculate the average number of trials until the third success, you divide 1 by the probability of success:
1 / 0.30 = 3.33
Therefore, on average, you would have to draw from the population about 3.33 individuals to get three subjects with the trait.
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A mall distribution center processes as many as 150,000 pieces of mail each day. The mail is sent via ground and air. Each land carrier takes 2000 pieces per load and each air carrier 1500 pieces per load. The loading equipment is able to handle as many as 200 loads per day.
Let x be the number of loads by land carriers and y the number of loads by air carriers. Which system of inequalities represents the situation?
Options:
1500x+2000y>=150,000 x+y>=200
2000x+1500y>=150,000 x+y>=200
2000x+1500y<=150,000 x+y<=200
1500x+2000y<=150,000 x+y<=200 ...?
Answers
is the correct option
Answer: [tex]2000x+1500y\leq150,000[/tex]
[tex]\Rightarrow x+\leq200[/tex]
Step-by-step explanation:
Let x be the number of loads by land carriers and y be the number of loads by air carriers.
Given: A mall distribution center processes as many as 150,000 pieces of mail each day.
Each land carrier takes 2000 pieces per load and each air carrier 1500 pieces per load.
[tex]2000x+1500y\leq150,000[/tex]
The loading equipment is able to handle as many as 200 loads per day.
[tex]\Rightarrow x+\leq200[/tex]
Hence, the system of inequalities represents the situation is
[tex]2000x+1500y\leq150,000[/tex]
[tex]\Rightarrow x+\leq200[/tex]
Find the lowest common denominator for the set of fractions: 6/a^2-7a+6 and 3/a^2-36
Answers
Answer:
(a-6)(a-1)(a+6)
Final answer:
The lowest common denominator for the fractions [tex]6/(a^2-7a+6)[/tex] and [tex]3/(a^2-36)[/tex] is (a-6)(a-1)(a+6).
Explanation:
To find the lowest common denominator for the fractions [tex]6/(a^2-7a+6)[/tex] and [tex]3/(a^2-36),[/tex] we need to determine the least common multiple of the denominators.
The denominator of the first fraction can be factored as (a-6)(a-1) and the denominator of the second fraction can be factored as (a-6)(a+6).
The common factors are (a-6) and (a-1)(a+6).
Therefore, the lowest common denominator is (a-6)(a-1)(a+6).
AT THANKSGIVING DINNER, EMMA'S FAMILY ATE 1/3 OF A PUMPKIN PIE. WHAT FRACTION OF THE PIE WAS LEFT OVER?
Answers
A fraction is written in the form of a numerator and a denominator where the denominator is greater than the numerator.
Total pumpkin in fraction = 1
= 1 - 1/3
= 2/3
The fraction of the pie left over is 2/3
What is a fraction?A fraction is written in the form of a numerator and a denominator where the denominator is greater than the numerator.
Example: 1/2, 1/3 is a fraction.
We have,
Amount of pumpkin pie eaten by the family = 1/3
Let the total pumpkin in fraction = 1
Now,
The fraction of the pie left over.
= 1 - 1/3
= (3 -1)/ 3
= 2/3
Thus,
The fraction of the pie left over is 2/3
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Suppose
cos(π/2−x) =3/5, cos x =4/5
find sin, tan, csc, sec, cot
Answers
To find the values of sin, tan, csc, sec, and cot, we use trigonometric identities and given information. Sin x = 3/5, tan x = 3/4, csc x = 5/3, sec x = 5/4, and cot x = 4/3.
Explanation:To find the values of sin, tan, csc, sec, and cot, we can use the trigonometric identities. From the given information, we know that cos x = 4/5. Using the Pythagorean identity, sin² x = 1 - cos² x, we can find sin x as:
sin x = sqrt(1 - (4/5)²) = sqrt(1 - 16/25) = sqrt(9/25) = 3/5
Now, we can calculate the remaining trigonometric values:
tan x = sin x / cos x = (3/5) / (4/5) = 3/4
csc x = 1 / sin x = 1 / (3/5) = 5/3
sec x = 1 / cos x = 1 / (4/5) = 5/4
cot x = 1 / tan x = 1 / (3/4) = 4/3
The first term in a sequence is x. Each subsequent term is three less than twice thw preceding term. what is the 5th term in the sequence?
A) 8x-21
B) 8x-15
C) 16x-39
D) 16x-45
E) 32x-93 ...?
Answers
x, 2x-3,
2(2x-3)-3,
2(2(2x-3)-3)-3,
2(2(2(2x-3)-3)-3)-3
So the answer is D) 16x-45.
Five boxes of candles contain a total of 60 candles. each box holds the same number of candles.
Answers
Answer:
60/5=12
Step-by-step explanation:
What is the solution to the equation -2n -2 / 3 =12 ? n =
Answers
If m arc AB = 120°, then which of the following have measures of 60°? (pick as many as you want)
a)∠ADB
b)∠ACD
c)∠BCA
d)∠BAD
e) m arc BC
Answers
The correct answers are:
a)∠ADB; and c)∠BCA.
Explanation:
The measure of an intercepted arc is twice as much as the measure of its inscribed angle. AB corresponds with both ∠ADB and ∠BCA. Since the arc is 120°, each angle would be half of that, or 60°.
Which law would you use to simplify the expression (x^⁴)⁹ ?
A. product of powers
B. power of a product
C. power of a quotient
D. power of a power
Answers
D. power of a power
x^⁴)⁹ = (x^4*9)= x^36
if you see this similar format you will recognize the equation as power of a power ^^•
Which of the following is a factor of 3x3 + 18x2 + 27x?
9x
x3
x + 3
x - 3
Answers
3x (x^2+6x+9)
3x (x+3). (x+3)
3x. (x+3)^2
3x (x+3)(x+3). Hope this is the answer that you are looking for.
An electrician charges a $50 house call fee and $65 per hour for work
Answers
A=65x + 50
A= Answer
X = number of hours
The radius,
R
, of a sphere is
5.8m
. Calculate the sphere's volume,
V
.
Use the value
3.14
for
π
, and round your answer to the nearest tenth.
Answers
(4/3)*π*5.83≈817.28323 or Rounded to the nearest tenth, 817.3
a student finds the slope of a line between (14,1) and (18,17). she writes 1-17/18-14 what mistake did she make?
Answers
Answer:
Mistake was in denominator.
Step-by-step explanation:
If a line passes through two points ( a, b ) & ( c, d )
Then slope of this line will be = [tex](\frac{d-b}{c-a})[/tex]
Now we come to the question.
a student finds the slope of a line passing through ( 14, 1 ) and ( 18, 17 ) as
slope = [tex](\frac{1 - 17}{18-14})[/tex]
This formula is not the form of [tex](\frac{d-b}{c-a})[/tex]
Accurate answer should be
slope = [tex](\frac{1-17}{14-18})[/tex]
So mistake was in denominator.
2-5|5m-5|=-73 absoute value
Answers
Point H is the incenter of triangle ABC. Find DH.
a. 14
b. 7
c. 18
d. 21
Answers
Due to a lack of information provided, the length of segment DH in the triangle ABC, where H is the incenter, cannot be determined as provided options could all be possible or might be something entirely different.
Explanation:The question doesn't provide enough information to determine the length of segment DH. In a triangle, the incenter is the center of the inscribed circle (incircle). This is the point where all the angle bisectors of the triangle meet. Segment DH would be a line from the incenter to a point on one of the triangle's sides, in other words, a radius of the incircle. However, without more information such as the length of the sides of the triangle or its area, it isn't possible to determine the length of DH. It could be any of the provided options (14, 7, 18, 21) or something else entirely.
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The incenter of a triangle is the point where the angle bisectors of the three angles of the triangle intersect.
Explanation:The incenter of a triangle is the point where the angle bisectors of the three angles of the triangle intersect. To find the distance DH, we can use the fact that the incenter is equidistant from the three sides of the triangle.
Let AD, BE, and CF be the angle bisectors of triangle ABC, where D, E, and F are the points of intersection with the sides.The incenter H is equidistant from each side of the triangle. So, we have DH = EH = FH.Let DH = x. Then, we have AH = 2x, BH = 2x, and CH = 2x.Therefore, DH = x = (AH + BH + CH)/3 = (2x + 2x + 2x)/3 = 6x/3 = 2x.
Since DH = 2x, the distance DH will be twice the distance from the incenter to any side of the triangle. Without knowing the specific values of the sides of triangle ABC, we cannot determine the exact value of DH. Therefore, none of the given options (a, b, c, d) is the correct answer.
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A basketball player is 15 ft horizontally from the center of the basket, which is 10 ft off the ground. At what angle should the player aim the ball if it is thrown from a height of 8.2 ft with a speed of 29 ft/s ?
Answers
sin (theta) = (10 - 8.2)/ 15 = 3 / 25
this angle is the ratio of the vertical displacement of the projectile and the horizontal displacement of the projectile.time taken to reach the top most point can found out by by using the formula v = u + at in the vertical direction
intial velocity in vertical direction = 29 sin (theta)
final velocity in vertical direction = 0
acceleration in vertcal direction = - 32 ft / s^2 so u will get the time to be 29 sin (theta) / 32
vertical displacement = [29 sin (theta)] [(29 sin ( theta)/32)] - (1/2)32 [(29 sin (theta)/32)]^ 2
horizontal displacement = [29 cos ( theta )] * 29 sin (theta) / 32
vertical displacement / horizontal displacement= 3/25
if u substitute the values u will get tan (theta) = 6 / 25
The Shelby University rose garden is a rectangular shape with an area of 192 square feet. The dimensions of the garden are x^2 feet wide by x^2– 61 feet long. How wide is the garden?
a. 3 feet c. 64 feet
b. 8 feet d. 131 feet
...?
Answers
x² = 64
(64 - 61) x 64 = 192
SERIOUSLY NEED HELP WITH THIS QUESTION.
The cube in the image has a volume of 1,000 cubic feet. The other solid has the same base and height as the cube, but the length of each of its slanted sides is 2 units longer than the height. What is the volume of the tilted solid?
800 cubic feet
1,000 cubic feet
1,200 cubic feet
2,000 cubic feet
Answers
Imagine you cut cube 2 peaces so that your cut is normal to bases and it passes throught both bases. If you take those 2 pieces and switch their position and stick one to another you will get non tilted original cube.
what you can conclude from this is that tilting doesnt influence volume if bases and height remain the same.
Answer:
The correct option is 2.
Step-by-step explanation:
It is given that the volume of cube is 1,000 cubic feet.
The volume of cube is
[tex]V=a^3[/tex]
[tex]1000=a^3[/tex]
[tex]a=10[/tex]
The side length of cube is 10 feet.
It is given that the other solid has the same base and height as the cube, but the length of each of its slanted sides is 2 units longer than the height.
The area of tilted solid is
[tex]V=\text{Area of base}\times \text{height}[/tex]
Since height and base are same, so the area of tilted solid is
[tex]V=(10 \times 10)\times 10=1000[/tex]
The volume of the tilted solid is 1,000 cubic feet. Therefore the correct option is 2.
What is 8/25 as a percentage?
Answers
Describe the graph of the function f(x) = x3 − 11x2 + 36x − 36. Include the y-intercept, x-intercepts, and the shape of the graph.
Answers
2) x-intercept => y = 0 => factor the function (start by dividing by x -2)
f(x) = (x-2)(x-3)(x-6) =0 => x =2, x = 3, x = 6 (these are the x-intercepts)
3) critical points:
between x = 2 and x = 3, there is a local maximum
between x =3 and x = 6 there is a local minimum
3) Shape.
The function comes growing from - infinity.
In the third quadrant the function is negative (it does not pass throuhg the second quadrant)
It enters to the fourth quadrant intercepting the y-axis at y = -36. It continues growing and intercepts the x-axis at x = 2.
It continues increasing until a maximum local positive value, starts to decrease, intercepts the x-axis at x = 3, continues decreasing, becomes negative, gets a local minimum in the fourth quadrant, starts to increase, intercepts the x-axis at x = 6, becomes positive, and continues growing.
Philip charges $12 to rake the yard she charges $5 per hour to bag the piles of leaves for 2 hours what is the expression
Answers
Point E has a negative x-coordinate, and its y-coordinate is not 0
Where could point E be located on the coordinate plane?
Answers
45 equals the quotient of a number n and 3
Answers
The equation representing the student's question is 45 = n / 3. To find n, we multiply both sides by 3, resulting in n = 135.
Explanation:The student's question, "45 equals the quotient of a number n and 3," can be translated into a mathematical equation. The word 'quotient' indicates a division operation. We set up the equation as 45 = n / 3.
To find the value of n, we multiply both sides of the equation by 3, giving us 3 * 45 = n. Therefore, n equals 135. The solution to this problem demonstrates how to manipulate an equation to solve for an unknown variable.
In this question, 45 is equal to the quotient of a number, represented by n, and 3.
We can represent this equation as:
45 = n / 3
To find the value of n, we can multiply both sides of the equation by 3:
45 * 3 = n
Therefore, n is equal to 135.
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what is the distance between 0 and 7/10?
Answers
Given the set of numbers (3, 5, 7, 11, 23), what number must be added to the set so that the median of the 6 numbers is 9? what number must be added to the set so that the mean of the 6 numbers is 9?
Answers
median is the number in the middle when they are arranged in increasing numerical order
example
1,2,3
median is 2 becaus it is in middle
1,4,6,7,8
6 is median
if you have an even amount then average the middle two
1,3,5,7
median is the mean of 3 and 5 which is 4
so
3,5,7,11,23
median is 7 right now
there are an odd amout of numbers
adding 1 will result in even
to find the median of even set of numbers, average the 2 middle ones
right now 7 is center
the mean of 5 and 7 is 6
the mean of 7 and 11 is 9
we want a mean of 9
so add any number greater than or equal to 11
3,5,7,11,12,23 is one option
3,5,7,11,23,50 is another
mean is 9
mean is the average of them
x is the number to be added
6 numbers will be the set since there are 5 now and adding 1 will get 6
(3+5+7+11+23+x)/6=9
(49+x)/6=9
times both sides by 6
49+x=54
minus 49 both sides
x=5
the number added must be 5
to make the median of the 6 numbers 9, add any number greater than or equal to 11, such as 12 or 234
to make the mean 9, add 5 to the set
Evaluate 0!÷1!.
0
1
undefined
Answers
Answer:
it is actually not undefined because of the (!), due to this the answer is 1
Step-by-step explanation:
0! 1 5! 120
1! 1 6! 720
2! 2 7! 5,040
3! 6 8! 40,320
4! 24 9! 362,880
those are the factorial calculations
A first number plus twice a second number is 13 twice the first number plus the second totals 20 find the numbers
Answers
A + 2B = 13
2A + B = 20
Solve these equations. You will get the equation
3B = 6. Therefore, B = 2.
Put this into either of the equations to find that A = 9. You can then check by putting A = 9 and B = 2 into both the equations to see if it is true. In this case, it is. Therefore, the solution is A = 9, B = 2.
A = 9, B = 2.
Find the coordinates for the vertex of the graph of the function y = 3x^2- 12x + 12
Answers
3(x²-4x+4).
now do bottoms up method.
1 × 4= 4, and -4 is the middle number.
-2 and -2 multiply to 4 and add up to -4.
factor: 3(x-2)(x-2). vertex form 3(x-2)²
vetex coordinates: (2,0)
If you pay for $25.00 for purchase which includes 11% sales tax. how much is the sale tax?
Answers
= $2.50
1% = $25.00 / 100
$0.25
Add them together to get 11%.
$0.25 + $2.50 = $2.75
Please make this the brainiest answer :3.
Sales tax is $ 2.75
Further Explanation
How to calculate
Tax = $ 25.00 x (11/100)
Tax = $ 25.00 x 0.11
Tax = $ 2.75
Initial purchase of $ 22.25 + 11% Tax ($ 2.75) = $ 25.00
Or other alternatives:
11% tax
n = sales tax value
11% = $ 25.00 / n
n = $ 25.00 x 11%
n = ($ 25.00 x 11) / 100
n = $ 275/100
n = $ 2.75
So, the sales tax is $ 2.75
Sales tax (VAT) is tax before Value Added Tax (VAT) and is charged each time a sales transaction. A VAT is charged at the manufacturer level / not to retailers (end users). A VAT is collected when delivering goods or services.
Definition of Tax Debt
Debt is an engagement as a result of a special agreement called a debt payable, which requires the debtor to pay the amount of money he has borrowed from creditors.
Sales tax / value-added debt or Sales tax payable is the company's debt to the tax office for sales tax collected by the company from customers for the sale of goods/services. Sales tax rates are deposited by the tax office multiplied by sales / net sales (net sales).
Example
$ 12 cash sale including 10% VAT
Tax Amount: 10 / ÷ 100 × 12 = $ 1.2
Sales Amount: 12 - 1.2 = $ 10.8
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Details
Class: High School
Subject: Mathematics
Keywords: VAT, Tax, Sales