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Ivahew [28]
3 years ago
8

Carter drawers one side of equilateral triangle pqr on the coordinate plane at points P(-3, 2) and Q(5, 2). Which ordered pair i

s a possible coordinate of vertex R?
Mathematics
1 answer:
Neko [114]3 years ago
7 0

Given: Two coordinates of an equilateral triangle P(-3, 2) and Q(5, 2).

To find: Coordinate of third vertex R.

Solution: Let us take coordinate of R is (x,y).

Because we are given PQR is an equilateral triangle, all side of the triangle PQR are equal.

We can say PQ = QR = RS.

Let us find equations by using distance formula.

Distance between two coordinates is given by formula

Distance = \sqrt{(x2-x1)^2+(y2-y1)^2}

PQ=\sqrt{(5-(-3))^2+(2-2)^2}  = \sqrt{(5+3)^2+(0)^2} = \sqrt{8^2+0} = \sqrt{64} =8

QR=\sqrt{(x-5)^2+(y-2)^2}  = \sqrt{x^2-10x+25 + y^2-4y+4} = \sqrt{x^2+y^2-10x-4y+29}

RS=\sqrt{(x-(-3)^2+(y-2)^2}  = \sqrt{x^2+6x+9 + y^2-4y+4} = \sqrt{x^2+y^2+6x-4y+13}.

We know, PQ= QR and PQ= RS.

\sqrt{x^2+y^2-10x-4y+29}[/tex] = 8

Squaring both sides, we get

x^2+y^2-10x-4y+29 =64         -------- equation (1)

and \sqrt{x^2+y^2+6x-4y+13}[/tex] = 8

Squaring both sides, we get  

x^2+y^2+6x-4y+13=64            ---------- equation (2).

Subtracting equation 2 from equation 1.

[/tex]x^2+y^2-10x-4y+29[/tex] = 64

[/tex]x^2+y^2+6x-4y+13[/tex] = 64

----------------------------------------------------

-16x +16 = 0

Subtracting 16 on both sides ,

-16x +16-16 = 0-16

-16x = -16

Dividing -16 on both sides.

-16x/-16 = -16/-16

x=1.

Plugging x=1 in first equation  

(1)^2+y^2-10(1)-4y+29 = 64

1+y^2-10-4y+29 = 64        (Simplifying)

y^2-4y+20 = 64

Subtracting 64 from both sides.

y^2-4y+20-64 = 64-64

y^2-4y-44 = 0

By using quadratic formula..

y=\frac{-(-4)+\sqrt{(-4)^2-4(1)(-44)} }{2(1)}  and

y=\frac{-(-4)-\sqrt{(-4)^2-4(1)(-44)} }{2(1)} [tex]y=2\left(1+2\sqrt{3}\right),\:y=2\left(1-2\sqrt{3}\right)

In decimal form

y=8.928, -4.928.

Therefore, coordinates of vertex R could be (1, 8.928) or (1, -4.928).


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Answer:

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Step-by-step explanation:

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In the second case, the parable curve is modeled by:

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we get:

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x₂ = 0.08/0.002

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3 years ago
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Okay. So I find it easiest to calculate the full area as if the shape was a rectangle, then subtracting open space. We already know that length is 100, so now we just need to find width. 50 + 25 + 50 + 25 + 50 = 200. 200 x 100 =20,000. So now we just need to find the areas of open space and subtract. Both spaces have a length of 75 and a width of 25. 25 x 75 = 1875. There were 2 spaces, and 1875 x 2 = 3,750. 20,000 - 3,750 = 16,250.


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bezimeni [28]

Answer:

(7x+6y) is a binomial factor of the given expression  49x^2+84xy+36y^2

Therefore  49x^2+84xy+36y^2=(7x+6y)(7x+6y)

Step-by-step explanation:

Given expression is 49x^2+84xy+36y^2

To find the binomial factor :

49x^2+84xy+36y^2

Rewritting the above expression as

=7^2x^2+2(7)(6)xy+6^2y^2

=(7x)^2+2(7x)(6y)+(6y)^2

=(7x+6y)^2 ( using the formula =(a+b)^2=a^2+2ab+b^2 here a=7x and y=6y )

=(7x+6y)(7x+6y) ( using the formula =(a+b)^2=(a+b)(a+b) )

Therefore  49x^2+84xy+36y^2=(7x+6y)(7x+6y)

(7x+6y) is a binomial factor of the given expression  49x^2+84xy+36y^2

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