Prove: the midpoint of the hypotenuse of a right triangle is equidistant from three vertices
Answers
From the picture attached,
Coordinates of point A → (0, 0)
Coordinates of point M → [tex](\frac{a}{2}, \frac{b}{2} )[/tex]
Coordinates of point A → (0, b)
Coordinates of point A → (a, 0)
Since, we have to prove that midpoint (M) of the hypotenuse BC is equidistant from all three vertices A, B and C,
AM = BM = CM
Use the expression for the distance between two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].
Distance = [tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
AM = [tex]\sqrt{(\frac{a}{2}-0 )^2+(\frac{b}{2}-0)^2}[/tex]
= [tex]\sqrt{(\frac{a}{2})^2+(\frac{b}{2} )^2}[/tex]
BM = [tex]\sqrt{(a-\frac{a}{2} )^2+(\frac{b}{2})^2}[/tex]
= [tex]\sqrt{(\frac{a}{2})^2+(\frac{b}{2})^2}[/tex]
CM = [tex]\sqrt{(0-\frac{a}{2} )^2+(b-\frac{b}{2})^2}[/tex]
= [tex]\sqrt{(\frac{a}{2})^2+(\frac{b}{2})^2}[/tex]
Therefore, AM = BM = CM = [tex]\sqrt{(\frac{a}{2})^2+(\frac{b}{2})^2}[/tex]
Learn more,
https://brainly.com/question/2520846
YES, the midpoint of the hypotenuse of a right triangle is equidistant from three vertices -
AM = CM = BM.
We have a right - angled triangle → ΔABC.
We have to prove that midpoint of the hypotenuse of a right triangle is equidistant from three vertices.
What is Right Angled Triangle?A right angled triangle is a triangle with one of the angles as 90 degrees.
According to the question -
Consider the hypotenuse BC. The mid - point of BC has the coordinates as -
M([tex]\frac{a}{2} , \frac{b}{2}[/tex]).
Now -
Length AM →
AM = [tex]\sqrt{(\frac{a}{2}) ^{2} + (\frac{b}{2}) ^{2} }[/tex]
Length CM →
CM = [tex]\sqrt{(\frac{a}{2}) ^{2} + (\frac{b}{2} - b) ^{2} }=\sqrt{(\frac{a}{2}) ^{2} + (\frac{b}{2}) ^{2} }[/tex]
Length BM →
BM = [tex]\sqrt{(\frac{a}{2} -a) ^{2} + (\frac{b}{2} ) ^{2} }=\sqrt{(\frac{a}{2}) ^{2} + (\frac{b}{2}) ^{2} }[/tex]
Therefore -
AM = CM = BM
Hence, the midpoint of the hypotenuse of a right triangle is equidistant from three vertices.
To solve more questions on Right angled triangle, visit the link below-
https://brainly.com/question/14260489
#SPJ2
Related Questions
How many grams are in 40 hectograms
Answers
There are 4000 grams in 40 hectograms.
To convert from hectograms to grams, you need to know that there are 100 grams in 1 hectogram.
Therefore, to find out how many grams are in 40 hectograms, you can use the conversion factor:
40 hectograms * 100 grams/hectogram = 4000 grams
So, there are 4000 grams in 40 hectograms.
To learn more about `hectograms; -
brainly.com/question/31122800
#SPJ6
Can a cubic equation have 3 complex roots? 3 real roots? ...?
Answers
Yes, a cubic equation can have three real roots if it cuts the x-axis three times.
Final answer:
A cubic equation can have either three real or three complex roots (including a real root and a pair of complex conjugate roots), determined by its discriminant. Complex roots arise in pairs due to the conjugate root theorem, ensuring that the coefficients of the equation remain real.
Explanation:
A cubic equation, which is an equation of the form ax^3 + bx^2 + cx + d = 0, can indeed have 3 complex roots or 3 real roots, depending on the discriminant and the nature of the coefficients. According to the Fundamental Theorem of Algebra, every cubic equation has three roots in the complex number system, which includes real numbers as a subset.
If the discriminant is positive, the cubic equation has three distinct real roots. If the discriminant is zero, the equation has a multiple root and all its roots are real.
Equilibrium problems in mathematics may require using cube roots when finding the solution, and utilizing the correct functions on a calculator is vital for these calculations. This could be in applied sciences like chemistry or physics, where cubic equations may arise.
Gina works at a diner. She earns $6 each hour plus tips.In one week,she worked 37 hours a week and earned $43 in tips.How much did she make altogether? Write an equation. Use a variable for the unknown.Solve.
Answers
The equation is x=6h+6
Greatest common factor of 12, 60, and 68
Answers
12: 2, 2, 3
60: 2, 2, 3 5
68: 2,2,17
Now you circle the numbers that are the same in each.
GCF: 2, 2
Multiply these numbers to find the GCF: 4
Use the substitution method to solve the system of equations. 2x - y = 10 3x - 2y = 8
Answers
3x-2y=8
-y= -2x +10 multiply by(-1)
y=2x-10
3x-2(2x-10)=8
3x-4x+20=8
-x=-12 multply by (-1)
x=12
24-y =10
-y=-14
(12,14)
Answer:
x=12, y=14
Step-by-step explanation:
[tex]2x - y = 10[/tex]
Solve the equation for y. Subtract 2x from both sides
[tex]- y = -2x+10[/tex]
Divide both sides by -1
[tex]y=2x-10[/tex]
Substitute 2x-10 for y in the second equation
[tex]3x - 2(2x-10)= 8[/tex]
[tex]3x - 4x+20= 8[/tex]
[tex]-1x+20= 8[/tex]
Subtract 20 from both sides
[tex]-1x= -12[/tex]
Divide both sides by -1
x= 12
[tex]y=2x-10[/tex]
Substitute 12 for x
[tex]y=2(12)-10=14[/tex]
y=14
The length of a rectangle is 3 meters more than twice its width. the length of this rectangle is 10 meters. write an equation to solve the perimeter of this rectangle
Answers
Eric sells shirts. It costs him $100 to buy all the shirts and $10 to print designs on the shirts. Eric sells each shirt for $4. So far Eric has sold 20 shirts. Eric uses an equation to calculate his total profit.
What is the slope of the equation modeling this situation?
Answers
To find the slope of the equation modeling Eric's situation, subtract the total cost from the selling price per shirt, resulting in a loss of $106 per shirt sold.
The slope of the equation modeling Eric's situation can be calculated as follows:
Profit per shirt = Selling price per shirt - Total cost per shirtProfit per shirt = $4 - ($100 + $10)Profit per shirt = $4 - $110 = -$106The slope, which represents the profit per shirt, is -$106, indicating a loss of $106 for each shirt sold.
Carolyn is building a pen for her dog. The perimeter of the pen is 52 ft
a) Explain how you can use a proportion to convert the perimeter into yards and feet.
b) The fencing material is sold by the yard. It costs 10.99 per yard. What is the cost of material before taxes
Answers
b)52/3=17 1/3 17 1/3*10.99= 190.493333333...(you get the point)
Carolyn's dog pen with a 52 ft perimeter is approximately 17.33 yards in length. Fencing material costs $10.99 per yard, making the total cost before taxes around $190.44.
Carolyn is constructing a dog pen with a perimeter of 52 feet. To convert the perimeter from feet to yards, we can use the fact that 1 yard equals 3 feet. Since she is purchasing fencing material by the yard, Carolyn needs to know the total length of fencing in yards. This requires setting up a proportion:
1 yard / 3 feet = x yards / 52 feet
Multiplying both sides by 52 gives us x = 52 / 3 yards. Upon calculating, we find that the pen's perimeter is approximately 17.33 yards.
Since the fencing material costs $10.99 per yard, to find the total cost before taxes, we multiply the number of yards needed by the price per yard:
17.33 yards × $10.99/yard = $190.4367
The cost for the fencing materials before taxes is approximately $190.44.
a magazine costs 3.50 how much can you buy with 100$
Answers
A pack with 6 cans of juice costs 3.48. What it the unit price?
A.0.49 per can B.0.50 per can C.0.58 per can D.0.60 per can**my answer??**
Answers
The unit price of one can of juice, when a 6-pack costs $3.48, is $0.58 per can.
The question asks us to calculate the unit price of a juice can when a pack of 6 cans costs $3.48.
To find the unit price, we divide the total cost of the pack by the number of cans in a pack.
In this case, we divide $3.48 by 6, which gives us $0.58.
$3.48 / 6 = $0.58
Therefore, the unit price per can is $0.58.
4x + 5x + 12 = -6 (given)
9x + 12 = -6 (simplify)
9x = -18 (subtraction)
x = -2 (addition)
Which reason is incorrect?
A) addition
B) given
C) simplify
D) subtraction
Answers
So the answer is choice A) addition.
Answer:
Addition
Step-by-step explanation:
USA Testprep
each month for 7 months, samuel mows 3 lawns. how many more lawns does he need to mow before he has mowed 29 lawns?
Answers
$800,000 if your rate of return is 5.3% compounded monthly
If you deposit the amount you need to achieve your goal in 20 years, how much will your savings be worth after 10 years?
Answers
What is the domain of the quadratic function graphed below?
Answers
36% of what number is 180?
Answers
36% of x = 180
36/100×x=180
x=(180×100)/36
x=500
What is the equation in point slope form of the line that passes through the point (2, 6) and has a slope of 5?
y−2=5(x−6)
y+2=5(x+6)
y−6=5(x−2)
y+6=5(x+2)
Answers
Answer:
The answer is Y-6=5(x-2)
a dragongfly traveled at a rate of 35 miles per hour for 2.5 hours .what distance did the dragonfly travel?
Answers
Here, Speed = 35 mi/hr
Time = 2.5 hr
Substitute their values into the expression,
Distance = 35 * 2.5 = 87.5
So, your final answer is 87.5 miles
Hope this helps!
In the diagram below, BC is an altitude of ABD. To the nearest whole unit, what is the length of CD?
Answers
Answer:
Actually the answer is 56, so D.
Step-by-step explanation:
To the nearest whole unit, the length of CD is approximately 4 units.
Given Information:
In triangle ABD, BC is an altitude (perpendicular) from point B to side AD.
We need to find the length of CD.
Approach:
Since BC is an altitude, triangle BCD is a right triangle with BC as one of its legs and BD as its hypotenuse.
We can use the Pythagorean theorem to relate the lengths of BC, CD, and BD.
Pythagorean Theorem: The Pythagorean theorem states that in a right triangle:[tex][ \text{Hypotenuse}^2 = \text{Leg}_1^2 + \text{Leg}_2^2 ][/tex]
Application to Triangle BCD:
Let’s denote the length of CD as (x).
BC is given as 30 units.
BD can be expressed as (BD = AB + AD), where AB is the length of the other leg.
Using Triangle ABC:
Triangle ABC is also a right triangle.
We have AC = 16 units (given).
Using the Pythagorean theorem for triangle ABC: [tex][ AB^2 + BC^2 = AC^2 \ AB^2 + 30^2 = 16^2 \ AB^2 + 900 = 256 \ AB^2 = 256 - 900 \ AB^2 = -644 ][/tex]
Since we cannot have a negative side length, we discard this solution.
Using Similar Triangles:
Notice that triangle BCD is mathematically similar to triangle ACE.
The coefficient of similarity is evident from the side CE.
We can set up the following proportion: [tex][ \frac{BC}{CD} = \frac{AC}{CE} \ \frac{30}{x} = \frac{16}{CE} \ CE = \frac{16x}{30} ][/tex]
Finding BD:
We know that (BD = AB + AD).
Since we couldn’t find AB directly, we express it in terms of CE:[tex][ AB = AC - CE = 16 - \frac{16x}{30} = \frac{480 - 16x}{30} ][/tex]
Now, we can find BD: [tex][ BD = AB + AD = \frac{480 - 16x}{30} + 16 = \frac{480 + 480 - 16x}{30} = \frac{960 - 16x}{30} ][/tex]
Applying the Pythagorean Theorem to Triangle BCD: [tex][ BD^2 = BC^2 + CD^2 \ \left(\frac{960 - 16x}{30}\right)^2 = 30^2 + x^2 \ 960^2 - 32x \cdot 960 + 256x^2 = 900x^2 + 900 \cdot 30^2 \ 960^2 - 32x \cdot 960 = 900x^2 - 900 \cdot 30^2 \ 960^2 + 900 \cdot 30^2 = 932x^2 \ x^2 = \frac{960^2 + 900 \cdot 30^2}{932} \ x \approx 3.75 ][/tex]
To the nearest whole unit, the length of CD is approximately 4 units.
Rewrite with only sin x and cos x.
cos 3x
cos x - 4 cos x sin2x
-sin3x + 2 sin x cos x
-sin2x + 2 sin x cos x
2 sin2x cos x - 2 sin x cos x
Answers
Final answer:
To rewrite expressions using only sin x and cos x, we can use trigonometric identities like [tex]cos 2x = 1 - 2sin^2 x[/tex] and sin 2x = 2sin x cos x. By applying these identities, we can simplify the given expressions.
Explanation:
To rewrite the given expressions using only sin x and cos x, we can use the following trigonometric identities:
[tex]cos 2x = 1 - 2sin^2 x[/tex]sin 2x = 2sin x cos x[tex]sin 3x = 3sin x - 4sin^3 x[/tex]Using these identities, we can rewrite the expressions as follows:
cos 3x = [tex]4cos^3 x - 3cos x[/tex]cos x - 4cos x sin^2 x = cos x - 4sin x cos x sin x = [tex]cos x - 4sin^3 x cos x[/tex]-sin 3x + 2sin x cos x = [tex]-(4sin^3 x) + 2sin x cos x[/tex] = [tex]-4sin^3 x + 2sin x cos x[/tex]-sin 2x + 2sin x cos x = -2sin x cos x + 2sin x cos x = 0[tex]2sin^2 x cos x - 2sin x cos x[/tex] = 2sin x cos x (sin x - 1)A donut shop is experimenting with two new types of cream-filled treats, one with raspberry filling, and the other with peach. A batch of raspberry donuts cooks in 11 minutes while a batch of peach donuts cooks in only 8 minutes.
Let x represent the number of raspberry-filled donuts and let y represent the number of peach-filled donuts. Write an inequality to represent how many batches of the new type of donuts can be produced in a 10 hour day.
Answers
11x < or = 600
8y < or = 600
or combine them:
11x + 8y < or = 600
Answer:
The required inequality is [tex]11x + 8y \leq 600[/tex]
Step-by-step explanation:
Given : A donut shop is experimenting with two new types of cream-filled treats, one with raspberry filling, and the other with peach.
A batch of raspberry donuts cooks in 11 minutes while a batch of peach donuts cooks in only 8 minutes.
Let x represent the number of raspberry-filled donuts and let y represent the number of peach-filled donuts.
To write : An inequality to represent how many batches of the new type of donuts can be produced in a 10 hour day.
Solution :
Time taken to cook the raspberry donuts = [tex]11x[/tex] minutes
Time taken to cook the peach donuts = [tex]8y[/tex] minutes
Donuts produced in 10 hours
1 hours = 60 minutes
10 hours = 60 × 10 = 600 minutes
Total time ≤ 600 minutes
Therefore, the required equation is
→ [tex]11x + 8y \leq 600[/tex]
7x-5y=4; what does x and y equal
Answers
7x - 5y = 4 Add 5y to both sides
7x = 5y + 4 Divide both sides by 7
x = [tex] \frac{5}{7} [/tex]y + [tex] \frac{4}{7} [/tex]
or
x = [tex] \frac{5y + 4}{7} [/tex]
To solve for y:
7x - 5y = 4 Subtract 7x from both sides
-5y = -7x + 4 Divide both sides by -5
y = [tex] \frac{7}{5} [/tex]x - [tex] \frac{4}{5} [/tex]
or
y = [tex] \frac{7x - 4}{5} [/tex]
the ordered pair for an x-intercept has an x-value of 0 A.True or False
Answers
You question basically is saying- (0,a number)
If we wanted the x intercept it would have to be (a number,0)
The ordered pair for an x-intercept has an x-value of 0 has some value of x and it has zero value for y, So it is a False statement.
What is the equation of line?The equation of a straight line is a relationship between x and y coordinates, The general equation of a straight line is y = mx + c, where m is the slope of the line and c is the y-intercept.
Given statement,
the ordered pair for an x-intercept has an x-value of 0
we can see in graph that when line intersect the x axis, the x coordinate of that point is known as x-intercept,
suppose, we have one line , y = 3x+3
so when we draw graph for it,
and if we see x-intercept
it will come out as, 0 = 3x+3 ⇒ x = -1
It is clear that value of x is not zero
hence, the given statement is false
To know more about Line check:
https://brainly.com/question/17188072
#SPJ2
The seats at a local baseball stadium are arranged so that each row has five more seats than the row in front of it. If there are four seats in the first row, how many total seats are in the first 24 rows?
These are the choices
1,476
2,856
2,952
1,428 ...?
Answers
The area of a rectangle painiting is given by the trinomial x^2-9x-36. what are the possible dimensions of the painting ? use factoring
Answers
The first term is, x2 its coefficient is 1 .
The middle term is, -9x its coefficient is -9 .
The last term, "the constant", is -36
Step-1 : Multiply the coefficient of the first term by the constant 1 • -36 = -36
Step-2 : Find two factors of -36 whose sum equals the coefficient of the middle term, which is -9 .
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -12 and 3
x2 - 12x + 3x - 36
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-12)
Add up the last 2 terms, pulling out common factors :
3 • (x-12)
Step-5 : Add up the four terms of step 4 :
(x+3) • (x-12)
Which is the desired factorization
what is 3/20 as a decimal equivalent
Answers
PLEASE HELP BEGGGING YOU
the point Q(9,-1) is reflected across the y axis. Use arrow notation to describe the original point and its reflection
Q(9,-1)----Q(9,1)
Q(9,-1)----Q (-9,1)
Q(9,-1) ------Q(-9,-1)
Q(9,-1) ----Q(9,-1)
Answers
Answer:
C is the answer.
Step-by-step explanation:
When you reflect over the y-axis, the y-axis will remain the same and the x-axis will change from either positive to negative, or negative to positive in this case it is positive to negative.
Sketch each parabola using the given information.
vertex (3, 6), y-intercept 2
Answers
put a point at (0,2)
hmm
parabola, so it is same on left and right
so from (3,6) to (0,2) is 3 down and 4 left
so we do the oposite in the other direction (left to right flip)
frmo (3,6), go 3 down and 4 right
get (3,10)
graph that pont as well
graph points (3,6) and (0,2) and (3,10)
(just fyi, the equaton is y=(-4/9)(x-3)^2+6)
One number is 3 less than twice another. If their sum is 39, find the numbers.
Which of the following systems of equations represents the word problem?
A.y = 2x - 3 and y = x + 39
B.y = 2x - 3 and x + y = 39
C.y = 2(x - 3) and x + y = 39 ...?
Answers
Let
y------> one number
x-----> another number
we know that
[tex]y=2x-3[/tex] ------> equation A
[tex]x+y=39[/tex] ------> equation B
substitute equation A in equation B
[tex]x+[2x-3]=39[/tex]
[tex]3x=39+3[/tex]
[tex]x=42/3=14[/tex]
Find the value of y
[tex]y=2*14-3=25[/tex]
the numbers are [tex]25[/tex] and [tex]14[/tex]
therefore
the answer is the option B
[tex]y=2x-3[/tex] and [tex]x+y=39[/tex]
Imaginary Numbers
Simplify
i^9
Answers
This means that i^2 = -1
If you square i until the 9th power, you will realize that i^9 = 1, or simply i.
So, your answer is i^9 = i.
Answer:
i^9 = i
Step-by-step explanation:
What is 2x-16=90 in Algebra1
Answers
90 + 16 = 106, then you divide by 2 to find x, and x equals 53.
step 1: add 16 to 90.
90 + 16= 106..
step 2: divide both side by 2.
106/2 = 53. 2x/2 would cancel out.
x= 53, and this is how you would solve to find x.