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We have a triangle JKL with midpoints M, N, and O of its sides. Then, we have another triangle PQR formed by the midpoints of the sides of triangle MNO.

The area of triangle PQR is given as 12. We need to find the area of triangle JQR.

Key Concept

Midpoint Theorem: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and half its length.  

Area Scaling: If the sides of a triangle are scaled by a factor of 'k', then its area is scaled by a factor of 'k^2'.

Solution

Triangle MNO and Triangle JKL:

Since M, N, and O are midpoints of JK, KL, and LJ respectively, triangle MNO is similar to triangle JKL with a scale factor of 1/2 (due to the midpoint theorem).

Therefore, the area of triangle MNO is (1/2)^2 = 1/4 times the area of triangle JKL.

Triangle PQR and Triangle MNO:

Similarly, triangle PQR is similar to triangle MNO with a scale factor of 1/2.

So, the area of triangle MNO is (1/2)^2 = 1/4 times the area of triangle PQR.

Finding the area of triangle JKL:

We know the area of triangle PQR is 12.

So, the area of triangle MNO is 12 * 4 = 48.

Therefore, the area of triangle JKL is 48 * 4 = 192.

Finding the area of triangle JQR:

Triangle JQR is half of triangle JKL (since QR is a median).

So, the area of triangle JQR is 192 / 2 = 96.

Therefore, the area of triangle JQR is 96.

Here is how can we compute the answer:

f(f(f(f(1+i) = f(f(f((1+i)^2) = f(f(f(0) = f(f(0) = f(0) = 0

Therefore, the answer is 0.

The answer is TP = 7*sqrt(3).

We're converting the power of a microwave from Watts (W) to a new unit, Zaps (z). We know the conversion rates for both Watts and Zaps to SI units (kg, m, s, min).

Watts to SI:

1 W = 1 kg * m^2 / s^3

Zaps to SI:

1 z = 1 kg * m^2 / min^3

We want to find the power of a 900 W microwave in Zaps. To do this, we can write an equality where the power is the same but expressed in different units:

900 W = P z

Now we can manipulate the equation to solve for P (power in Zaps). We can do this by introducing a conversion factor that equates Watts and Zaps.

This factor will cancel out the desired units (kg and m^2) and leave us with a factor that relates Watts and Zaps through time units (seconds and minutes).

a. We know from the definitions of Watts and Zaps that:

- 1 W / (1 kg * m^2 / s^3) = 1 z / (1 kg * m^2 / min^3)

b. This simplifies to:

- 1 W * (min^3 / s^3) = 1 z

c. This conversion factor is equal to 1 because we're converting between equivalent units that express the same fundamental quantities (mass, length, time) but in different time scales (seconds vs minutes).

Apply the conversion factor to the original equation:

900 W * (min^3 / s^3) = P z

Since the conversion factor is 1 (as derived previously), we have:

900 W = P z

Therefore, the power of the 900 W microwave in Zaps is also 900 Zaps. However, to express the answer in scientific notation, we should recognize that 900 can be written as 9.00 x 10^2.

Answer: The power of the 900 W microwave in Zaps is 9.00 x 10^2 Zaps.

We're converting the power of a microwave from Watts (W) to a new unit, Zaps (z). We know the conversion rates for both Watts and Zaps to SI units (kg, m, s, min).

Watts to SI:

1 W = 1 kg * m^2 / s^3

Zaps to SI:

1 z = 1 kg * m^2 / min^3

We want to find the power of a 900 W microwave in Zaps. To do this, we can write an equality where the power is the same but expressed in different units:

900 W = P z

Now we can manipulate the equation to solve for P (power in Zaps). We can do this by introducing a conversion factor that equates Watts and Zaps.

This factor will cancel out the desired units (kg and m^2) and leave us with a factor that relates Watts and Zaps through time units (seconds and minutes).

a. We know from the definitions of Watts and Zaps that:

- 1 W / (1 kg * m^2 / s^3) = 1 z / (1 kg * m^2 / min^3)

b. This simplifies to:

- 1 W * (min^3 / s^3) = 1 z

c. This conversion factor is equal to 1 because we're converting between equivalent units that express the same fundamental quantities (mass, length, time) but in different time scales (seconds vs minutes).

Apply the conversion factor to the original equation:

900 W * (min^3 / s^3) = P z

Since the conversion factor is 1 (as derived previously), we have:

900 W = P z

Therefore, the power of the 900 W microwave in Zaps is also 900 Zaps. However, to express the answer in scientific notation, we should recognize that 900 can be written as 9.00 x 10^2.

Answer: The power of the 900 W microwave in Zaps is 9.00 x 10^2 Zaps.

For problem 3, the answer is 3/2.