Trapezoid GHJK was rotated 180° about the origin to determine the location of G'H'J'K', as shown on the graph. What are the coordinates of pre-image point H?

The coordinates of pre-image point H are (3, 2).

Given that,

Trapezoid GHJK was rotated 180° about the origin to determine the location of G'H'J'K', as shown on the graph.

We have to determine,

What are the coordinates of pre-image point H?

According to the question,

Rotation in mathematics is a transformation that rotates a figure about a fixed point known as the center of rotation.

In rotation, there is a central point that remains fixed, and everything else rotates around that point.

The counterclockwise rotation of 180 degrees is centered at the origin. Again, as we did in the case of 90 degrees.

In the given diagram H' is located at (-3,-2).

By the rule for 180° rotation about the origin is :

(x, y) is equal to (-x, -y).

Then, the point H' located at (-3, -2) after rotation at 180 degrees the co-ordinate about the origin is,

(-3, -2) is equal to (3, 2).

Hence, the coordinates of pre-image point H are (3, 2).

For more details refer to the link given below.

link

H(3,2)

Step-by-step explanation:

The missing diagram is shown in the attachment.

The trapezoid GHJK was rotated 180° about the origin to determine the location of G'H'J'K', as shown on the graph.

We want to determine the coordinates of the pre-image point H.

From the diagram H' is located at (-3,-2).

We know the rule for 180° rotation about the origin is :

This means that:

Equating corresponding components, we have -x=-3.

This means that:

x=3.

Also, -y=-2, implies that y=2.

Therefore the coordinates of the pre-image point H is (3,2)

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4