The perimeter of triangle KBN is 61.3 units.

To find the perimeter, we will first need to find the lengths of the sides KB and BN.

By drawing a diagram and labeling the angles and sides, we can see that triangle KBV and CBV share the same base, and angles KBV and CBV are vertical angles, making them equal.

Similarly, angles CNA and CKA are equal, since they are vertical angles sharing the same base in triangles ACN and ACK.

Therefore, we can conclude that triangle KBV and triangle ACK are similar, since they have two angles that are equal.

Using this similarity, we can set up a proportion between the sides of the triangles:

KB/AC = BN/BC

Solving for KB, we get: KB = (BN * AC)/BC

Substituting the given values, we get KB = (18 * 25) /18 = 25 units.

Similarly, we can find BN by setting up a proportion between triangles BNV and ABC and solving for BN.

Using the Pythagorean theorem, we can also find the length of VN, which is 24 units.

Now, we have all the required lengths to find the perimeter of triangle KBN, which is:

Perimeter(KBN) = KB + BN + KN = 25 + 24 + 12.3 = 61.3 units

The side of a triangle is equal to 14, and the angles adjacent to it are 45 and 105 degrees. To find the side opposite to the angle in question, we can use the Law of Sines.

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In other words, a/Sin(A) = b/Sin(B) = c/Sin(C), where a, b, and c are the sides of the triangle and A, B, and C are the opposite angles.

Applying this to our problem, we can set up the following equation: 14/Sin(45°) = x/Sin(105°), where 14 is the known side, x is the unknown side, and 45° and 105° are the known adjacent angles.

Solving for x, we get x = 14 * Sin(105°)/Sin(45°) = 21.73. Therefore, the side opposite to the angle in question is approximately 21.73 units long.

Remember, always draw a diagram and label the sides and angles correctly when solving for unknowns in a triangle. It helps to visualize the problem and avoid confusion.

To find the degree measure of angle OMK, we need to use the angle sum property of a triangle. In this case, we can use the given information to find the missing angle.

First, we know that the sum of all the angles in a triangle is always 180 degrees. Therefore, we can write the following equation:

N 49° L + K + OMK = 180°

Simplifying, we get:

K + OMK = 180° - N 49° L

Now, we can use the given information that angle OMK is equal to angle K to rewrite the equation as:

K + K = 180° - N 49° L

Combining like terms, we get:

2K = 180° - N 49° L

Dividing both sides by 2, we get:

K = (180° - N 49° L) / 2

Finally, to find the degree measure of angle OMK, we just need to substitute the value of K in the equation K + OMK = 180° - N 49° L.

Therefore, the degree measure of angle OMK is:

OMK = (180° - N 49° L) / 2.

The equation of line BC is y = (1/7)x + 14/7.

The equation of the line containing the height BH of this triangle is y = -7x + 11. Keep in mind that the height BH is perpendicular to the base BC and passes through the point B. Hence, the slope of the height BH is the negative reciprocal of the slope of the base BC.

To find the slope of the base BC, use the slope formula (𝑦2−𝑦1/𝑥2−𝑥1) with the points B(8,2) and C(7,9).

(9-2)/(7-8) = -7

Since the slope of the base BC is -7, the slope of the height BH would be the negative reciprocal, which is 1/7.

To find the y-intercept of the height BH, substitute the coordinates of point B into the equation y = mx + b and solve for b.

2 = (1/7)*8 + b

b = 14/7

Hence, the equation of the line containing the height BH is y = (1/7)x + 14/7.

For the final step, input the desired value for x into the equation of the height BH to find the corresponding y-value. Remember, x corresponds to the x-coordinate of the point on the base BC, while y corresponds to the distance from that point to the height BH.

The area of triangle ABCD can be calculated using the formula:

Area = (perimeter/2) * inradius = (56/2) * (8 + (3+8)/2) = 28 * 11.5 = 322 square units.