You buy 3 shirts and 1 pair of pants for $32. Your friend buys 2 shirts and 3 pairs of pants for $40. How much did each item cost?
First let’s define a shirt as X and pants as Y!
In the first scenario, the equation will be 3X + Y =$32
In the second scenario, the equation will be 2X + 3Y = $40
Now because we have two different variables (X & Y) let’s solve for one of them (Ex: Y) to simplify the equations!
3X + Y= $32—> subtract 3X on both sides of the equation
-3X. - 3X
Net result: Y= 32- 3X
Now we can insert this into the second equation 2X + 3Y = $40 in order to solve for X variable
2X + 3(32-3X) = $40
2X + 96 - 9X = $40 —> simplify the equation
-7X + 96 = $40 —> substrate 96 from both sides
-7X = -56 —> divide both sides by -7
X= 8
Now reinsert 8= X into 3X + Y = $32
3(8) + Y = $32
24+ Y = $32 —> substract 24 from both sides
y = 8
ANSWER: X= 8, Y= 8
Jim began a 145 mile bicycle trip to build up stamina for a triathlon competition. Unfortunately, his bicycle chain broke, so he finished the trip walking. The whole trip took 7 hours. If Jim walks at a rate of 4 miles per hour and rides at 30 miles per hour, find the amount of time he spent on the bicycle.
Answer:
The amount of time he spent on the bicycle using substitution method is 4.5 hours .
Step-by-step explanation:
The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
A linear equation is defined as an equation that is written in the form of Ax+By=C. It is the combination of two variables and a constant value present in them.
The first step in the substitution method is to find the value of any one of the variables from one equation in terms of the other variable.
Let the total time taken to complete the race by bicycle be x
Let the total time taken to complete the race by walking be y
We know,
Speed=Distance/Time
4y+30x=145
x+y=7
Using the substitution method ,
x=7-y
4y+30(7-y)=145
4y+210-30y=145
210-145=26y
y=65/26
y=2.5
So, the amount of time he spent on bicycle is x
x=7-y
x=7-2.5
x=4.5 hours
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describe the effect of the transformation (x,y) to (x,y+4)
A. vertical stretch without reflection
B. horizontal translation of 4 units
C. vertical stretch with reflection
D. Vertical translation of 4 units
1. ABCD is a square with side length 12 cm.
a) Find the exact area of the shaded
[2 MARKS]
b) Find the area, correct to 2 decimal places. Use =3.142. [1 MARK]
region. [Leave your answer in terms of #]
NOT
A
6
с
I
Answer:
Exact Area of shaded region = 141.39cm²
Step-by-step explanation:
Quadrant of a circle - Quadrant refers to the four quarters in the coordinate plane system. Each of the four sections is called a quadrant. When it comes to circle, the quarter of a circle is called a quadrant, which is a sector of 90°.
area of circle =πr²
area of quadrant of a circle = πr²/4
where r is the radius of the circle.
In the given figure there are 2 types of quadrants of circle are present
one with radius AB (bigger )
another with radius AB/2 (smaller)
Now area of quadrant having radius = AB
⇒ π(AB) ²/4
here AB is the side of square
so AB = 12cm
substituting this in the area of quadrant formula
area of bigger quadrant = π(12²)/4
area₁ = 144×π/4
area₁ = 144×3.142/4
area₁ = 113.112cm²
now area of smaller quadrant = π(AB/2) ²/4
area ₂ = π(6) ²/4
area ₂ = 36π/4
area ₂ = 9×π
area ₂ = 28.278cm²
Now it is clear from the given figure,
area of shaded region is = area₁ - area₂ +2×area₂
= 113.112cm² - 28.278cm² + 2× 28.278cm²
= 113.112cm² + 28.278cm²
= 141.39cm² (answer)
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help please!!!!!!!!!
Determine the length
The length of KI is 10 cms.
What is Length?The skill of measuring the lengths of objects and understanding the units of length is very crucial as it helps us to interact with our environment in a more organized manner.
Here, ΔRJS ≈ Δ IJK
then JR / JI = SR/KI
1/2 =5 /ki
kI = 5 X 2
kI = 10 units.
Thus, The length of KI is 10 cms.
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If the rejection region is +1.10, what will be your decision if the z computed value is 2.34
The decision when the z value is within the rejection region is that we'll have to reject the null hypothesis.
What is a rejection region?It should be noted that the rejection region is also known as the critical region. It is the set of values where the null hypothesis is rejected.
When the observed test statistic is in the rejection region, we'll have to reject the null hypothesis and then accept rte alternative hypothesis.
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can disjunction introduction (either ∨il or ∨ir ) be used to derive any type of formula other than a disjunction?
It is impossible to use a disjunction introduction to derive any other type of formula besides disjunctive formulas.
What is Disjunction Introduction?A rule of inference in propositional logic and virtually every other deduction system is referred to as disjunction introduction.
It is right to state that it is not possible to use a disjunction introduction to derive any other type of formula besides disjunctive formulas.
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Please explain the bottom two the ones with the explanation box
First we must solve the bottom situations.
___________________________________________ (2A and 2B)
First expression (2A): 2v+8.3+2v-6.1
⇒ Collect like terms
(2v + 2v) + (8.3 - 6.1)
⇒ Simplify
4b + 2.2
Second expression (2B): 3v+14.4-v
⇒ Collect like terms
(3v - v) + 14.4
⇒ Simplify
2v + 14.4
They are not equivalent because when simplified they both have different coefficients and constants each of different values.
_____________________________________________ (4A and 4B)
First expression (4A): 10 + 3 (1/3x + 2)
⇒ Simplify 1/3x to x/3.
10 + 3(x/3 + 2)
⇒ Expand by distributing terms.
10 + x + 6
⇒ Collect like terms.
x + (10 + 6)
⇒ Simplify.
x + 16
Second expression: x + 16
This expression can not be simplified any further. This is because it is a simplified version of another expression.
These expressions are equivalent because expression 4B is the simplified version of expression 4A.
______________________________________
Level: 7th Grade
Unit: Distributing and Combing Like Terms
Determine which graph represents a reflection across the x-axis of f(x) = 3(1.5)x.
The reflection is written as:
[tex]g(x) = -3*(1.5)^x[/tex]
And the graph can be seen below.
How to get the graph of the reflection?
Here we have the function:
[tex]f(x) = 3*(1.5)^x[/tex]
A reflection across the x-axis is given by:
[tex]g(x) = -f(x).[/tex]
Then, replacing the function f(x) by the actual equation:
[tex]g(x) = -3*(1.5)^x[/tex]
The graph of the function g(x) can be seen below:
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15-
4
3+
2H
11
-5-4-3-2-1₁
Mark this and return
2345
-3-
2 3 4 5 x
What is the range of the function on the graph?
all the real numbers
all the real numbers greater than or equal to 0
all the real numbers greater than or equal to 2
all the real numbers greater than or equal to -3
The question was incomplete. Below you will find the missing graph image.
The range contain all the real numbers greater than or equal to –3 .
Range of function : The set of outputs of a function is called range of that function. Range of a function is also called co-domain of the function.
Example : If f(x) = x² is the function with domain as all real numbers, then the range of that function is the set of all non-negative real numbers.
We have to find the range of the function on the graph.
We can see by the graph minimum value of y of function is -3 and the graph continues to extend upward in the positive direction.
So, the range contain all the real numbers greater than or equal to –3 .
Therefore, the range contain all the real numbers greater than or equal to –3 .
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x(y − 3) + n(3 − y)
Consider the quadratic function y = 1/5 (x – 2)2 + 5.
Which statements are true about the function and its graph? Select TWO options.
A) The vertex of the function is (–2, 5).
B) There are no real roots for the function.
C) The graph of the function opens down.
D) The graph contains the point (2, 5).
E) The graph intersects the x-axis at one unique point.
Answer:
B and D
Step-by-step explanation:
vertex(2,5) so D is correct
there are no real roots for the function B is correct
To support a shelf, the diameter of a bolt can measure anywhere from 0.2 cm less than 1.5 cm to 0.2 cm greater than 1.5 cm. Which graph represents the possible measures of the diameter of the bolt?
a shape is rotated 180 to its origin and then its image s reflected in the y axis describe fully the translation that would have the same result as the two transformation drscribe above
The single transformation that will cause the same effect as that of these two transformation is the reflection of shape in the x-axis .
(x,y) -----------> (x,−y)
What is Transformation ?The change done to a shape on a coordinate plane by reflection ,translation and rotation is a called Transformation .
It is given that a shape is rotated to its origin
Then its image is reflected in the y axis.
The single transformation that would have the same result as the two transformations is (x,y) to (x,−y)
reflection of shape in x-axis .
The x stays the same, but the y changes sign.)
The change caused by Rotation by 180° about the origin,
(x,y) -------> (−x,−y)
The change caused when reflected in the y-axis,
(x,y) --------> (−x,y)
Therefore the single transformation that will cause the same effect as that of these two transformation is the reflection of shape in the x-axis .
(x,y) -----------> (x,−y)
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Suppose that the weight (in pound) of an airplane is a linear function of the amount of fuel (In gallons) in it's tank. When carrying 20 gallons of fuel, the airplane weighs 2210 pounds. When carrying 54 gallons of fuel, it weighs 2397 pounds. How much does the airplane weigh if it's carrying 60 gallons of fuel?
Answer:
For 60 gallons of fuel, the airplane would weigh 2430 gallons.
Step-by-step explanation:
Concept: Use the values and conditions to make a linear equation of a straight line. Let the y-axis be fuel and x-axis be airplane weight. The given points are (2210, 20) and (2397, 54). Use these points to make an equation of line.
[tex]\frac{y-20}{x-2210} =\frac{54-20}{2397-2210} \\\frac{y-20}{x-2210} = \frac{34}{187}[/tex]
Let this be the equation of straight line.
We have to find the airplane weight at 60 gallons of fuel. This means y = 60 and we have to find the x.
[tex]\frac{60-20}{x-2210} =\frac{34}{187} \\x-2210=\frac{(187)(40)}{34} \\x-2210=220\\x=2430[/tex]
So, the airplane weight is 2430 gallons.
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The hollow sphere, in which the circus motorcyclist performs his stunts, has a diameter of 7 m. Find the area available to the motorcyclist for riding.
[tex]\huge\color{red} {\boxed{\tt{A}}} \color{blue} {\boxed{\tt{N}}} \color{hotpink}{\boxed{\tt{S}}} \color{blueviolet}{\boxed{\tt{W}}} \color{fuchsia}{\boxed{\tt{E}}}\color{lavender}{\boxed{\tt{R}}}[/tex]
[tex] \red{ \star} \pmb{Diameter \: of \: the \: sphere = 7 m}[/tex]
[tex] \red{ \star} \pmb{Radius =\frac{7}{2} = 3.5 m}[/tex]
[tex] \pink{\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }}[/tex]
[tex] \sf{Area \: available \: to \: motorcyclist \: for \: riding = Area \: of \: sphere}[/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= 4πr^2}[/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= 4*\frac{22}{7}*3.5*3.5}[/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= 154}[/tex]
The area available to the motorcyclist for riding is 154 m²
Diameter of the sphere = 7m
[tex]{Radius =\frac{7}{2} = 3.5 m}[/tex]
Area available to motorcyclist for riding = Area of sphere
[tex]{= 4πr^2} \\ \\ {= 4*\frac{22}{7}*3.5*3.5} \\ \\ {= 154}.[/tex]
The area available to the motorcyclist for riding is 154 m²
On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 3, negative 7) and (0, 2). Everything to the left of the line is shaded.
Which linear inequality is represented by the graph?
y < 3x + 2
y > 3x + 2
y < One-thirdx + 2
y > One-thirdx + 2
The correct answer is option B which is y > 3x + 2.
What is inequality?Inequality is the relationship between two expressions that are not equal, employing a sign such as ≠ “not equal to,” > “greater than,” or < “less than.”.
The given statement says that On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 3, negative 7) and (0, 2)
The plot of the inequality is given in the graph attached with the answer the inequality will be y > 3x +2 which passes through the points (-3,-7 ) and ( 0, 2 ).
Therefore the correct answer is option B which is y > 3x + 2.
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In the Kite ABCD, AB = 5√2mm AP= 5mm, PD = 7mm, find the area to the nearest mm2.
The area of the kite ABCD will be 60 square millimeters.
What is Geometry?It deals with the size of geometry, region, and density of the different forms both 2D and 3D.
In the Kite ABCD, AB = 5√2mm = 7.07 mm, AP= 5mm, PD = 7mm
Then the area of the kite ABCD will be
Area = 2(area of triangle APB and area of triangle APD)
The side length of PB will be
AB² = AP² + PB²
PB² = AB² – AP²
PB² = 7.07² – 5²
PB² = 50 – 25
PB² = 25
PB = 5 mm
Then the area will be
Area = 2[(1/2 × 7 × 5) + (1/2 × 5 × 5)]
Area = 60 square mm
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Which equation justifies why 93 - ?
=
(0³) 1_9(²+³)_9
93
(³)
93
³_9(²+³)=9
(0³) -
93
93
=9=9
_g(¹-³)_9
=9
=9
The equation that justifies [tex]9^{1/3}=\sqrt[3]{9}[/tex] is [tex](9^{1/3}) ^{3} = 9^{3/3} = 9[/tex]
How to find equation that justifies another?The expression given is [tex]9^{1/3}=\sqrt[3]{9}[/tex].
The equation that justifies [tex]9^{1/3}=\sqrt[3]{9}[/tex] is as follows:
Therefore, using indices law,
[tex]a^{x/y} =\sqrt[y]{a^{x} }[/tex]
Hence,
[tex]9^{1/3}=\sqrt[3]{9}[/tex]
Therefore, the equation that shows it is correct is as follows;
[tex](9^{1/3}) ^{3} = 9^{3/3} = 9[/tex]
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DRAW AND WRITE SHORT NOTES ON THE FOLLOWING Properties of the various types 1 Triangle: (Equilateral, Isoceles, Right - angled, Scalene
2. Quadrilaterals. (square, rectangle, Rhombus, parallelograph trapezium, kite
ASAP PLEASE FIRST AND CORRECT WILL GRT BRAINLIEST ANSWER
Answer:
Triangles:
- Equilateral: Has three equal sides
- Isosceles: Has two equal sides
- Right angled: One angle of a right angle triangle measures 90 degrees
- Scalene: All sides are different lengths
Quadrilaterals:
- Square: Has four right angles and four equal sides
- Rectangle: Has four right angles and opposite sides are equal
- Rhombus: A parallelogram with four equal sides
- Parallelogram: Opposite sides are equal and parallel
- Trapezium: At least one set of sides are parallel
- Kite: Two pairs of adjacent sides are the same length
please be quick im being timed
Answer:
this one
Step-by-step explanation:
how many solutions exists for the given equation?
3x+13=3(x+6)+1
Answer:
No Solutions.
Step-by-step explanation:
original equation
3x+13 = 3(x+6)+1
distribute 3
3x+13 = 3x+18+1
simplify
3x+13 = 3x+19
since the slope is the same for both equations and the y-intercept is different the two equations are parallel meaning they will never intersect so there are no solutions.
Answer:
No solutions
Step-by-step explanation:
3x + 13 = 3 (x + 6) + 1
3x + 13 = 3x + 18 + 1
3x + 13 = 3x + 19
3x + 13 = 3x + 19 meaning no solution since they aren't equivalent to each other
does someone mind helping me with this problem? Thank you!
Answer:
-20
2(xy-7)=-20 with x=-1 y=3
Step-by-step explanation:
2(xy-7)=?
When variables are connected such as xy, this means to multiply.
x=-1
y=3
Multiply the x and the y together.
-1*3=-3. Plug that into the equation.
2(-3-7)=?
Subtract the inside the parenthesis.
2(-10)=?
Multiply. Remember when multiplying a positive and a negative the answer will always be a negative.
2(-10)=-20
This means that 2(xy-7)=-20 with the x=-1 and y=3.
Hope this helps!
If not, I am sorry.
In the diagram, a circle centered at the origin, a right triangle, and the Pythagorean theorem are used to derive the equation of a circle, x2 + y2 = r2.
On a coordinate plane, circle Q is centered at the origin with radius r. Triangle P Q S is shown. Point Q is at (0, 0) and points P is at (x, y). The length of Q P is r, the length of P S is y, and the length of Q S is x. Angle P S Q is a right angle.
If the center of the circle were moved from the origin to the point (h, k) and point P at (x, y) remains on the edge of the circle, which could represent the equation of the new circle?
(h + x)2 + (k + y)2 = r2
(x – h)2 + (y – k)2 = r2
(k + x)2 + (h + y)2 = r2
(x – k)2 + (y – h)2 = r2 7-foot radius.
Based on the information provided, the equation of this new circle is equal to: B. (x - h)² + (y - k)² = r²
The equation of a circle.Mathematically, the equation of a circle can be derived by using Pythagorean theorem;
x² + y² = r²
Where:
h and k represents the coordinates at the center.r is the radius of a circle.After translating h units along the x-axis and k units along y-axis on a coordinate plane, the equation of this new circle is given by:
(x - h)² + (y - k)² = r²
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Answer: B) (x – h)2 + (y – k)2 = r2
Step-by-step explanation:
What is the standard form of 0.00849
Answer:
8.49 × 10⁻³
Explanation:
Standard form of a number: A. × 10ᵇ
(where 'A' is the number, b is the exponent over 10)
For the number "0.00849", move the decimal point 3 places right so the exponent of 10 is -3.
Standard form: 8.49 × 10⁻³
Other examples of standard/scientific forms are: standard forms
i) 0.043, move the decimal point 2 places right = 4.3 × 10⁻²
ii) 4030, move the decimal point 3 places left = 4.03 × 10³
Answer:
when a number is given in that manner you have to take that decimal between two whole numbers.
and that is the main reason why I took that decimal from 0.0 to 8.49 and since it went towards the right my answer is going to be 8.49 *10 raised to negative 3.
The total number of degrees in the center is 360°. if all five vertex angles meeting at the center are congruent, what is the measure of a base angle of one of the triangles? 54° 72° 108° 144°
Answer:
54°
Step-by-step explanation:
Since the 5 vertex angles are congruent, each vertex angle = 360/5 = 72°
The sum of the angles of each triangle is 180-72 = 108°
The base angles of each triangle are equal to each other so each base angle = 108/2 = 54°
PLEASE HELP!
Find the degree measure of each numbered angle.
Answer:
1=76°
2=76°
3=36°
Step-by-step explanation:
To find 1, add the other angles in the triangle and subtract by 180.
85+19=104
180-104=76.
1 is equal to 2 so 2=76
To find 3, add 76 and 68, then subtract by 180.
76+68=144
180+144=36
3=36
Hope this helps!
If not, I am sorry.
Answer:
m∠1 = 76°
m∠2 = 76°
m∠3 = 36°
Step-by-step explanation:
The interior angles in a triangle sum to 180°
⇒ m∠1 + 85° + 19° = 180°
⇒ m∠1 + 104° = 180°
⇒ m∠1 + 104° - 104° = 180° - 104°
⇒ m∠1 = 76°
Vertical Angle Theorem
When two straight lines intersect, they form two pairs of angles. The vertically opposite (non-adjacent) angles are congruent.
⇒ m∠2 = m∠1
⇒ m∠2 = 76°
The interior angles in a triangle sum to 180°
⇒ m∠2 + m∠3 + 68° = 180°
⇒ 76° + m∠3 + 68° = 180°
⇒ m∠3 + 144° = 180°
⇒ m∠3 + 144° - 144°= 180° - 144°
⇒ m∠3 = 36°
If you vertically stretch the exponential function f(x)=2^x by a factor of 3 what is the equation of the new function
The new equation will become f(x) = 3([tex]2^{x}[/tex]).
The given information is the stretch factor which is 3 and the equation f(x)=[tex]2^{x}[/tex].
When there is a vertical strecth factor, you need to multiply the equation by the stretch factor.
By stretching the function, the value of y has to be multiplied by 3, will now be equal to 3f(x)
The new equation will become f(x) = 3([tex]2^{x}[/tex]).
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I want to verify my work answer for the below SSAT question.
Thank you
Cindy throws a rock in the air. The rock's height, y, in feet, with respect to time, x, in seconds can be modeled by the function y=-2x^2+10x+28. When does the rock hit the ground in seconds
My work:
y=-2x^2+10x+28
0= -2(x^2-5x-14) (While knowing that the zero is supposed to be there, I don't understand why 0= -2(x^2-5x-14) )
(x-7)(x+2)
(-7,+2)
I put 2 as the answer, but when checking it was 7?
What part did I do wrong?
Answer:
7
Step-by-step explanation:
You were correct in setting the equation equal to 0 because the problem is asking for when the rock hits the ground and in this problem, you may assume that the ground is y=0, unless the problem says otherwise.
Ok so where you went wrong is that when you factored out and got
(x-7)(x+2), which is correct, you forgot to set these equal to zero (because your actual equation is set to zero). So you should have
x-7 = 0
and
x+2 = 0
and this gives the actual answers +7 and -2. And since time cant be negative its 7 seconds.
This was a silly mistake and happens a lot more often than you think, so I wouldn't worry too much but make sure you practice more, better to lose points on harder questions than easy ones.
Good luck with your exam !
Answer:
7 seconds
Step-by-step explanation:
Given function:
[tex]y=-2x^2+10x+28[/tex]
where:
y = height above the ground (in feet)x = time (in seconds)To find the time when the rock hits the ground, set the equation to zero and solve for x.
First, simplify by factoring out the common term -2:
[tex]\implies -2(x^2-5x-14)=0[/tex]
Divide both sides by -2:
[tex]\implies x^2-5x-14=0[/tex]
To factor a quadratic in the form [tex]ax^2+bx+c[/tex], find two numbers that multiply to ac and sum to b.
[tex]\implies ac=1 \cdot -14=-14[/tex]
[tex]\implies b=-5[/tex]
Therefore, the two numbers that multiply to -14 and sum to -5 are: -7 and 2
Rewrite b as the sum of these two numbers:
[tex]\implies x^2-7x+2x-14=0[/tex]
Factorize the first two terms and the last two terms separately:
[tex]\implies x(x-7)+2(x-7)=0[/tex]
Factor out the common term (x - 7):
[tex]\implies (x+2)(x-7)=0[/tex]
Therefore:
[tex]\implies x+2=0 \implies x=-2[/tex]
[tex]\implies x-7=0 \implies x=7[/tex]
As time cannot be negative, the rock hits the ground in 7 seconds.
Select all the correct answers.
The diagram shows the construction of two tangent lines to a circle from a point outside the circle.
A circle with a center O. Two tangents, JP and KP are drawn from the same external point P. A horizontal line is drawn from P to O. A vertical line is drawn through the tangent and intersects the horizontal line at M. A line connects JM.
From the diagram, which three statements are true?
- line segment MJ≅ Line segment MP
-∡OJP≅∡OML
-∡OJM≅∡JMO
- line segment OM≅line segment MP
- line segment OJ≅line segment MJ
- line segment LM≅line segment MP
1) line segment MJ≅ Line segment MP
2) ∡OJP≅∡OML
3) line segment OM≅line segment MP
The below mentioned statements are true according to the figure , Option A, B and D is the correct answer.
What are Tangents ?A line that intersects the curve at a single point at 90 degree is called a tangent.
In the given figure it can be seen that the
∡OJP≅∡OML (right angle)line segment OM≅line segment MP (as M is bisecting the line OP)line segment MJ≅ Line segment MPTherefore , Option A , B and D is the correct answer
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