Prove that a parallelogram is a square iff its diagonals are both congruent and perpendicular
Answers
The lengths of the sides are a and b.
The lengths of the diagonals are 2x and 2y.
Because the diagonals are both congruent and perpendicular, therefore there are two right triangles as shown.
Note that x = y.
Because x = y, each right triangle is isosceles and has the angles 90°, 45° and 45°.
From the Pythagorean theorem,
For one right triangle,
a² = x² + y² = x² + x² = 2x².
For the other right triangle,
b² = y² + x² = x² + x² = 2x²
Therefore
a² = b²
a = b
It follows that all sides of the parallelogram are equal and each angle is
45+45 = 90°
Therefore the parallelogram is a square.
Related Questions
Given:
AC = BC
∠3 = ∠4
Based on the given information and the algebraic and geometric properties presented or proven thus far, choose the congruence theorem that could be used to prove that triangle ACM is congruent to triangle BCM. If it is not possible to prove the triangles are congruent, choose "not possible".
Choices:
SSS
SAS
ASA
not possible
Answers
Answer:
SAS Postulate
Step-by-step explanation:
Find the value of x so that the line passing through (10, 5) and (x, 9) has a slope of -2.
Answers
(y2 - y1)/(x2-x1)
y2 = 9
y1 = 5
x2 = x
x1 = 10
So:
(9-5)/(x-10) = -2
Now, we just solve for x,
9-5 = -2(x-10)
4 = -2x+20
2x = 16
x = 8
Find the 31st term of the following sequence.
9, 15, 21, ...
Answers
Notice that the above sequence is arithmetic because the common difference: d is constant.
d = second term - first term.
= 15 - 9
= 6
The general term of an arithmetic sequence is,
a_{n} =a_{1} +(n-1)d
Where , nth term = [tex] a_{n} [/tex]
[tex] a_{1} [/tex] = First term
and d = common difference.
The given sequence is: 9, 15, 21, ...
Here a_{1} = 9,
d = 6
We need to find the 31st term. So, n = 31.
Next step is to plug in these values in the above formula. Therefore,
[tex] a_{31} =9 +(31-1)*6 [/tex]
= 9 + 30* 6
= 9 + 180
= 189
So, 31st term of this sequence is 189.
Hope this helps you.
Answer:
189
Step-by-step explanation:
Two integers with the sum of -17 and the product of 52?
Answers
you have x+y=-17 ---------(1)
xy=52 ---------(2)
from eqn (1) x=-17-y --------(3)
, so subsitute eqn (3) in to eqn(2)
(-17-y)y=52
-17y-y²=52
:
-y²-17y-52=0 same as y²+17y+52=0
y²+13y+4y+52=0
y(y+13)+4(y+13)=0
(y+4)(y+13)=0
therefore y=-4 or -13
now substitute y=-4 or -13 in eqn (3)
x=-17-(-4) or -17-(-13)
x= -13 or -4
so when x=-13 ,y=-4 and when x=-4 ,y=-13
so the integers are -13 and -4
An automobile manufacturing organization noted a drop in sales after an incident occurred due to the lack of a safety feature in their cars. Many customers began considering other brands of cars after this incident. How can the organization avoid losing customers
Answers
Recalling the effected car and fixing the issue
Answer:
introducing customer retention strategies
Step-by-step explanation:
Which ordered pair comes from the table ? HELPPP!
Answers
apply the distributive property and the greatest common factor to write an equivalent expression.
32k + 24 = _ (_)
what numbers should be in place of the underscores?
Answers
Find three consecutive odd integers whose sum is negative 465
Answers
465 / 3 = 155
so -153 + -155 + -157 = -465
Find the 20th term in arithmetic sequence -4, 1, 6, 11, 16,
Answers
Jesse ate 100 donughts i 5 days each day she ate 6 more then the day before how many did she eat each day
Answers
hoped it helped! :D
Which postulate or theorem proves that these two triangles are congruent?
Answers
Answer:
option B
Step-by-step explanation:
Given in the question are ∠x ≅ ∠y and ∠zwx ≅ ∠ ywz and wz is common in both.
These angles are adjacent angles of triangles.
We know theorem for congruence is Angle - Angle - Side
Therefore, for the congruence of these two triangles option B, AAS congruence theorem is the answer.
does anyone know how to do this?
Answers
such that pq=12 and p+q=b
Assuming we're looking for simple values, you can start by factorizing 12:
12 = 3*4 = 2*6 = 12*1 = -3*-4 = -2*-6 = -12*-1
For 3 and 4, b would be 7
2 and 6 => b=8
12 and 1 => b=13
and for the negatives
b can be -7, -8 or -13
So the total set is -13, -8, -7, 7, 8, 13
There are many ways to do this.
X^2+bx+12=0
[tex](-b +- \sqrt{ b^{2} -4*12 } )/2 [/tex]
This equation gives a good answer if
[tex]b ^{2} \geq 48[/tex]
Which expression is a perfect cube?
A. -8x21y8
B. -64x64y64
C. -125x9y20
D. -216x9y18
Answers
Expression A:
[tex] \sqrt[3]{-8x^{21}y^{8}} [/tex]
[tex] \sqrt[3]{-8} [/tex] [tex] \sqrt[3]{x^{21}[tex] \sqrt[3]{y^8} [/tex]} [/tex]
[tex]-2x^ \frac{21}{3} y^ \frac{8}{3} [/tex]
[tex]-2 x^7 y^ \frac{8}{3} [/tex]
the term [tex]y^ \frac{8}{3} [/tex] is not a perfect cube, therefore expression A is not a perfect cube.
--------------------------------------------------------------------------------------------------------------
Expression B
[tex] \sqrt[3]{-64x^{64}y^{64}} [/tex]
[tex] \sqrt[3]{-64} \sqrt[3]{x^{64}} \sqrt[3]{y^{64}} [/tex]
[tex]-4 x^ \frac{64}{3} y^ \frac{64}{3} [/tex]
The term [tex]x^ \frac{64}{3} [/tex] and [tex]y^ \frac{64}{3} [/tex] are not perfect cube because 64/3 doesn't give whole number
----------------------------------------------------------------------------------------------------------------
Expression C
[tex] \sqrt[3] {-125x^9y^{20}} [/tex]
[tex] \sqrt[3]{-125} \sqrt[3]{x^9} \sqrt[3] {y^{20}} [/tex]
[tex]-5 (x^ \frac{9}{3}) (y^ \frac{20}{3})[/tex]
[tex]-5 (x^3) (y^ \frac{20} {3}) [/tex]
The term [tex]y^ \frac{20}{3} [/tex] are not perfect cube because 20/3 doesn't give whole number
---------------------------------------------------------------------------------------------------------------
Expression D
[tex] \sqrt[3]{ -216 x^{9} y^{18} } [/tex]
[tex] \sqrt[3]{-216} \sqrt[3]{x^9} \sqrt[3]{y^{18}} [/tex]
[tex]-6 ( x^ \frac{9} {3} ) ( y^ \frac{18} {3} ) [/tex]
[tex]-6 (x^3) (y^6)[/tex]
All terms are perfect cubes
ANSWER: OPTION D
The inequality x2 + 12x + 35 ≥ 0 has two critical points and three possible intervals for solutions. Choose each set of possible test points for the three intervals.
–8, –6, –4
–10, –6, 0
–6, 0, 6
–6, 0, 10
Answers
-8,-6,-4
-10,-6,0
these are the possible test points for your three intervals
The given inequality is, x² + 12 x+35≥0
→ x²+7 x +5 x+5×7≥0
→ x (x+7) +5(x+7)≥0
→ (x+5)(x+7)≥0
→ x+5=0 ∧ x+7=0 gives x= -7 ∧ x= -5
Now drawing the number line and marking point ,-5 and -7 on it.
Now the three intervals are (-∞ , -7], [-7,-5] and [-5,∞).
The solution set of inequality (x+5)(x+7)≥0 is (-∞ , -7] and [-5,∞).
The set of possible test points for
⇒ (-∞ , -7] → -8, -10
⇒ [-7,-5] → -6
⇒ [-5,∞) → -4, 0
Option (1) -8,-6,-4 and Option (2) -10,-6,0 satisfies the given condition.
which exponential functions have been simplified correctly check all that apply
Answers
Answer:
Option A, C, D, are the correct options.
Step-by-step explanation:
To check the exponential functions which have been simplified correctly we will take option one by one.
A). [tex]f(x)=5(\sqrt[3]{16})^{x}=5(\sqrt[3]{2^{4}})^{x}=5(2\sqrt[3]{2})^{x}[/tex]
B). [tex]f(x)=2.3(8^{\frac{1}{2}})^{x}=2.3(2.\sqrt{2})^{x}\neq 2.3(4^{x})[/tex]
C). [tex]f(x)=81^{\frac{x}{4}}=(3^{4})^{\frac{x}{4}}=(3^{\frac{4}{4}})^{x}=3^{x}[/tex]
D). [tex]f(x)=\frac{3}{4}(\sqrt{27})^{x}=\frac{3}{4}(\sqrt{3^{3}})^{x}=\frac{3}{4}.(3\sqrt{3})^{x}[/tex]
[tex]f(x)=(24)^{\frac{1}{3}x}=(2^{3}.3)^{\frac{1}{3}x}=(2.3^{\frac{1}{3}})^{x}\neq 2(\sqrt[3]{3})^{x}[/tex]
Exponential functions and their inverses like the natural logarithm are inverse functions, meaning they can undo each other. Simplifications like ln(e^x) result in x, and e^(ln x) simplifies to x. This property is integral to correctly simplifying exponential functions.
Explanation:The exponential functions and their inverses, like the natural logarithm (ln), have specific properties where they can "undo" each other. This relationship allows us to simplify expressions involving exponentials and logarithms. For example, when you have ln(ex), this simplifies to x because the natural logarithm and the exponential function are inverse functions.
Similarly, when applying the exponential function to a natural logarithm, like eln(x), the result is just x. Furthermore, when taking square roots of exponentials, you can adjust the exponent to be evenly divisible by 2 and then take the square root of the base number, followed by dividing the exponent by 2. Also, while dividing exponentials, you subtract the exponents when the bases are the same.
Here is an example of this simplification: to compute the square root of e4, we can write it as e4/2, which simplifies to e2.
If f(x)=5x^0+4x^-3 then f(a)=?
Answers
f(a)=5(a^0)+4a^-3 a^0=1
=5(1)+4/a^3
=5+4/(a^3)
To find f(a) for the given function, substitute a for x in the function and simplify the expression.
Explanation:Given the function f(x) = 5x0 + 4x-3, we need to find f(a). To find f(a), we substitute a for x in the function. Plugging in a, we get f(a) = 5a0 + 4a-3. Since any number raised to the power of 0 is equal to 1, we have f(a) = 5(1) + 4a-3. Simplifying further, we get f(a) = 5 + 4/a3.
Learn more about Functions here:https://brainly.com/question/21145944
#SPJ2
Given that a line has a slope of -3 and a point on the line is (2,-4),
1.what is the equation of the line in point-slope form?
2.how do we create slope-intercept form?
*please show all work, step by step.
Answers
y + 4 = -3(x - 2)
We can convert this to Slope-Intercept Form by isolating y.
y + 4 = -3(x - 2)
y + 4 = -3x + 6
Subbtract 4 from both sides.
y = -3x + 2
The equation in Slope-Intercept Form is y = -3x + 2.
PLZ HELP ILL GIVE A BRAINLIEST SHOW ALL WORK!!!!
Answers
(√3 + 2√3) + √7
Step 2. Simplify
3√3 + √7
That's all.
What’s the ratio of 42¢ to $1.25? (Change dollars to cents)
Answers
Answer:
42 125 becuase i got um right bozzzzzzzz
Step-by-step explanation:
What does it mean when u cant find the mean of a set of numbers?
Answers
Answer:
Another word for mean is the "average", your goal is to find the average of the following data set, it's pretty easy, you just have to add up all the numbers in the data set and divide it with the total numbers in the data set.
For example:
Find the mean of the following data set:
4, 1, 8, 5, 7, 3
Mean: 4 + 1 + 8 + 5 + 7 + 3 = 28
then we divide 28 with 6 since there are 6 numbers in the following data set.
Mean: 4 + 1 + 8 + 5 + 7 3 = 28/6 = 4.6
We get the final answer of 4.6!
Which postulate or theorem proves that △JHG and △EFG are congruent?
SAS Congruence Postulate
ASA Congruence Postulate
HL Congruence Theorem
AAS Congruence Theorem
Answers
The postulate that proves that △JHG and △EFG are congruent is;
Option B; ASA Congruence Postulate
From the given image, we can see two triangles namely;
△JHG and △EFG
Now, we can see that;
FG = GH
That is given because we see that G has divided FH into 2 equal parts.
Also, we see the right angle sign at ∠F and ∠H of both triangles.
Thus, since they are both right angles, then we can say that;
∠EFG = ∠JHG
Also, at point G, we have two vertically opposite angles which are;
∠EGF and ∠JGH
Now, from vertically opposite angle theorem, vertically opposite angles are always equal;
Thus; ∠EGF = ∠JGH
So far, we have;
FG = GH
∠EFG = ∠JHG
∠EGF = ∠JGH
This is 2 corresponding equal angles and 1 corresponding equal side. The side is included between the two angles and as such the proof that △JHG and △EFG are congruent is the ASA Congruence Postulate.
Read more at; https://brainly.com/question/19413244
Solve x ym = y x3 for m.
Answers
xym (1-1x2) = 0
xym (1 (un cuadrado)-1x2)=0
xym(1+x)(1-x)=0
1+x=0
1-x=0
m= 0
x=-1
x=1
Is true
m=0;x= -1;1
given a line whose slope is, m, is -6/5, and passes through point (-5,3), the graph will?
Answers
y₁ = 3; x₁ = -5; m = [tex]- \frac{6}{5} [/tex]
⇒ y - 3 = [tex]- \frac{6}{5} [/tex] ( x - (-5))
⇒ y - 3 = [tex]- \frac{6}{5} [/tex] (x + 5)
Now, you can approach here in two ways: put it in the y-intercept form or the general form.
In the y-intercept form;
y - 3 = [tex]- \frac{6}{5} [/tex]x + ([tex]- \frac{6}{5} [/tex] (5))
y - 3 = [tex]- \frac{6}{5} [/tex]x - 6
y = [tex]- \frac{6}{5} [/tex]x - 3
In the general form;
y - 3 = [tex]- \frac{6}{5} [/tex] (x + 5)
by multiplying through by 5
⇒ 5 ( y - 3 ) = -6 ( x + 5)
⇒ 5y - 15 = -6x - 30
⇒ 5y + 6x = -15
∴ the line that passes through (-5, 3), with gradient [tex]- \frac{6}{5} [/tex] is
y = [tex]- \frac{6}{5} [/tex]x - 3 (y-intercept form) OR 5y + 6x = -15 (general form).
Emily has 32 grams of a 25% sugar syrup, but this is just too sweet for her! How much pure water should she add to have a 10% syrup?
Answers
original water content= 32*75/100=24 grams
original sugar content= 32*25/100= 8grams
total mass at 10% syrup:
8grams/total mass=10%=0.1
total mass=8/0.1=80grams
total mass of water=80grams- 8grams of sugar content= 72 grams
total mass of water added=72grams- 24grams original water content= 48grams.
answer: 48grams
25%+0%=10%
32g xg (32+x)g
Now, we create an equation:
(0.25^32)+(0^x)=0.1^(32+x)
8+0=3.2+0.1x
8=3.2+0.1x
4.8=0.1x
48=x
Answer: 48 gram.
The lenght of a rectangleis 6 inches more thans its width the area of a rectangle is 91 inches find the dimernins
Answers
MATH PLEASE HELP 50 POINTS 1 QUESTION
Complete the reasons in the proof below.
Given: ∠1 and ∠5 are supplementary.
Prove: a ∥ b
Statements
1. ∠1 and ∠5 are supplementary
2. ∠1 ≅ ∠2
3. ∠2 and ∠5 are supplementary
4. a ∥ b
Answers
For 2, due to that 1 and 2 are vertical angles (angles that are opposite to each other when two lines cross according to Math is Fun)
For 3, since ∠1=∠2, ∠1 can be substituted by ∠2 in ∠1+∠5=180 to get ∠2+∠5=180. Therefore, they are supplementary.
For 4, due to that we have Consecutive Interior Angles (they are angles that are on one side of a transversal, which is the line that cuts through a and b,but inside the two lines according to Math is Fun) adding up to 180 degrees, a || b
Feel free to ask if you have any questions!
Each of ten tickets is marked with a different number from 1 to 10 and put in a box. if you draw a ticket from the box, what is the probability that you will draw 2, 8, or 4?
Answers
Answer: [tex]\dfrac{3}{10}[/tex]
Step-by-step explanation:
Given : Each of ten tickets is marked with a different number from 1 to 10 and put in a box.
i.e. The total number of outcomes : 10
Favorable outcomes : 2, 8, or 4
i.e. Number of favorable outcomes = 3
Now, the probability that you will draw 2, 8, or 4 is given by :-
[tex]=\dfrac{\text{Number of favorable outcomes}}{\text{Total outcomes}}\dfrac{3}{10}[/tex]
Please help! I need to show my work
Answers
Notice that sides AB and CD are horizontal, so they are parallel with different lengths.
Sides BC and Ad are not parallel.
A quadrilateral with two parallel sides and two non-parallel sides is a trapezoid.
The length of a rectangular plot of land is 10 yards more than its width. If the area of the land 600 square yards , find the dimensions
Answers
Final answer:
The dimensions of the rectangular plot of land, where the length is 10 yards more than its width and the area is 600 square yards, are 20 yards by 30 yards. We arrived at this by setting up a quadratic equation and solving for the width.
Explanation:
To find the dimensions of a rectangular plot of land where the length is 10 yards more than its width and the area is 600 square yards, we can set up an equation. Let w be the width of the plot. Then the length is w + 10 yards. The area of a rectangle is given by the formula length × width, which in our case is (w + 10) × w = 600.
Setting up the equation, we get:
w(w + 10) = 600
w^2 + 10w - 600 = 0
To solve the quadratic equation, we can factor it, use the quadratic formula, or complete the square. Factoring the equation, we find that:
(w - 20)(w + 30) = 0
Thus, the width w can be either 20 yards or -30 yards. Since a width cannot be negative, we take w = 20 yards. Then, the length is:
w + 10 = 20 + 10 = 30 yards
The dimensions of the rectangular plot of land are 20 yards by 30 yards.
Mrs. simms told her class to find two consecutive even integers such that twice the lesser of two integers is 4 less than two times the greater integer.
a. write and solve an equation to find the integers.
b. does the equation have one solution, no solutions, or is it an identity? explain.
Answers
Final answer:
To find two consecutive even integers, we can set up and solve an equation. The equation in this case is an identity, meaning it is true for all values of x. Therefore, there are infinitely many solutions.
Explanation:
To find two consecutive even integers, let's represent the first integer as x and the second integer as x + 2. Now, according to the given information, twice the lesser of the two integers is 4 less than two times the greater integer. We can write this as:
2x = 2(x + 2) - 4
Simplifying the equation:
2x = 2x + 4 - 4
2x = 2x
This equation is an identity, meaning it is true for all values of x. Therefore, there are infinitely many solutions. In this case, there are infinitely many pairs of consecutive even integers that satisfy the given conditions.