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Analyze Triangle Properties:

We are given that triangle ABC is a right triangle with angle A = 30 degrees and angle C = 90 degrees. This makes angle B = 60 degrees (since the angles in a triangle sum to 180 degrees).

Since BD bisects angle ABC, we know that angles ABD and CBD are both equal to half of angle ABC, which is 30 degrees.

Utilize Isosceles Triangle Property:

Because BD bisects angle B and is perpendicular to side AC (since AC is the hypotenuse of right triangle ABC), segment BD also bisects side AC. This makes triangles ABD and BDC congruent.

Find Lengths in Triangle ABD:

Knowing AC = 12 and that BD bisects AC, we can determine that AD = DC = 12 / 2 = 6.

Apply Area Formula for Triangle ABD:

The area of a triangle can be calculated using the formula: Area = 1/2 * base * height.

In triangle ABD, we know the base (AB) is part of the hypotenuse of right triangle ABC. Since angle A = 30 degrees, we can use the 30-60-90 right triangle properties to find that AB = 2 * AC = 2 * 12 = 24.

The height of triangle ABD is the perpendicular distance from point D to side AB. Since BD is perpendicular to AC, the height is the same length as segment BD. However, we haven't yet determined the length of BD.

Find BD using Pythagorean Theorem:

Since triangle ABD is a right triangle with AB = 24 and AD = 6, we can use the Pythagorean Theorem to find BD (the hypotenuse):

BD^2 = AB^2 - AD^2 BD^2 = 24^2 - 6^2 BD^2 = 576 - 36 BD^2 = 540 BD = √540 (Since we want the positive side length)

However, working with the square root of 540 might be cumbersome. We can simplify this by noticing that 540 can be factored as 2^2 * 3^3 * 5. Taking the square root of both sides:

BD = √(2^2 * 3^3 * 5) = 2 * 3 * √5

Calculate Area of Triangle ABD:

Now that we know all side lengths, we can find the area of triangle ABD:

Area = 1/2 * base * height

Area = 1/2 * 24 * (2 * 3 * √5)

Area = 72 * √5

Sure, let's break it down:

We're given triangle \(STU\) with midpoints \(M\), \(N\), and \(P\) on sides \(TU\), \(US\), and \(ST\) respectively. \(UZ\) is the altitude of the triangle.

Given:

- \(\angle TSU = 62^\circ\)

- \(\angle STU = 29^\circ\)

1. We draw the altitude \(UZ\) from vertex \(U\) to side \(TS\), creating right triangles \(TUZ\) and \(SUZ\).

2. We find:

   - \(\angle SUT = 180^\circ - 29^\circ - 62^\circ = 89^\circ\)

   - \(NM\) is parallel to \(ST\), so \(\angle NZM = \angle TSU = 62^\circ\)

   - \(NP\) is parallel to \(UT\), so \(\angle NPM = \angle STU = 29^\circ\)

3. Then, we calculate the sum:

   \[ \angle NZM + \angle NPM = 62^\circ + 29^\circ = 91^\circ \]

So, the sum of angles \( \angle NZM + \angle NPM \) is \( 91^\circ \).

We want to find the probability that the coin will be tossed more than $10$ times. We can achieve this in two scenarios:

After $10$ tosses, we have not yet seen three consecutive heads or three consecutive tails.

We have seen either three consecutive heads or three consecutive tails, but not until the $10$th toss.

Let's focus on the first scenario:

In this scenario, we need to find the number of valid sequences of $10$ tosses where neither three consecutive heads nor three consecutive tails occur.

Valid sequences include any combination of heads and tails as long as there are no three consecutive heads or three consecutive tails. For example: $HHTHHTHHTT$ or $THTHTHTHTH$.

Invalid sequences include: $HHHHHHHHHH$ (three consecutive heads), $TTTTTTTTTT$ (three consecutive tails), $HHTHHHHHHH$ (three consecutive heads).

There are $2^{10}$ total possible sequences of $10$ tosses.

Out of these, we need to subtract the number of sequences that have three consecutive heads or three consecutive tails.

The number of sequences containing three consecutive heads or three consecutive tails is $2^8$.

So, the number of valid sequences is $2^{10} - 2^8$.

The probability of each valid sequence is $\left(\frac{1}{2}\right)^{10}$.

Thus, the total probability of getting more than $10$ tosses is:

Total probability = (2^10 - 2^8)/2^10 = 3/4.

So, the probability that the coin will be tossed more than $10$ times is $\frac{3}{4}$.