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The proof is obvious if you use coordinates.

The answer is angle BPC is 20 degrees.

The sum is 8,514,402.

The 30th term is 50/7.

To solve this problem, we need to find the length \( QR \) given that the two circles are externally tangent and have radii of 8 and 12.

Let the centers of the two circles be denoted as \( O_1 \) and \( O_2 \), with \( O_1 \) being the center of the circle with radius 8 and \( O_2 \) being the center of the circle with radius 12. The distance between the centers \( O_1O_2 \) can be calculated as:

\[
O_1O_2 = r_1 + r_2 = 8 + 12 = 20.
\]

Next, we denote the lengths of the segments from the tangent point to the circle centers. Let \( d \) represent the distance between the circle centers, which we've calculated to be 20.

When a common external tangent is drawn, it intersects the line joining the centers of the two circles at a right angle at the point where it meets the external line \( PQ \). The formula for the length of the tangent \( t \) from an external point to a circle is given by:

\[
t = \sqrt{d^2 - (r_1 + r_2)^2},
\]

where \( d \) is the distance between the centers of the circles and \( r_1 \) and \( r_2 \) are the radii of the circles. However, here we use the tangent distance formula as follows:

\[
t^2 = d^2 - (r_1 - r_2)^2.
\]

Here, we have:

- \( r_1 = 8 \),
- \( r_2 = 12 \),
- \( d = 20 \).

Now we calculate \( r_2 - r_1 \):

\[
r_2 - r_1 = 12 - 8 = 4.
\]

Now substituting into the tangent formula:

\[
t^2 = d^2 - (r_2 - r_1)^2.
\]

Calculating \( d^2 \):

\[
d^2 = 20^2 = 400,
\]

and \( (r_2 - r_1)^2 \):

\[
(r_2 - r_1)^2 = 4^2 = 16.
\]

Substituting these values into the equation gives:

\[
t^2 = 400 - 16 = 384.
\]

Now, take the square root to find \( t \):

\[
t = \sqrt{384} = \sqrt{64 \times 6} = 8\sqrt{6}.
\]

Therefore, the length of the common tangent \( t \) is \( 8\sqrt{6} \).

Now, the problem requires finding \( QR \). Assuming that point \( R \) is the point where the tangent intersects line \( PQ \), and assuming line \( PQ \) is parallel to the line joining centers \( O_1 \) and \( O_2 \), we have that:

The distance \( QR \) from the tangent point to the line can often be represented relative to the tangent lengths, but specifics of \( QR \) in positions relative to other given points were not provided. If we are seeking \( QR \) as the length of the common external tangent, we state that:

\[
QR = 8\sqrt{6}.
\]

Thus, the final answer is:

\[
QR = 8\sqrt{6}.
\]

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.