To find the area of a triangle with given coordinates for the vertices, we can use the formula:

Area = (1/2) |(x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

Substituting the coordinates of the vertices A(2,4), B(8,4), and C(5,9) in the formula, we get:

Area = (1/2) |(2(4 - 9) + 8(9 - 4) + 5(4 - 4))|

Simplify:

Area = (1/2) |(-10 + 40 + 0)|
Area = (1/2) |30|
Area = 15

So, the area of the triangle ABC is 15. The correct answer is C) 15.

Complementary angles are angles whose measures add up to 90°. Let's call the measures of the two complementary angles 3x and 2x.

Thus, the equation can be written as:

3x + 2x = 90
5x = 90

Dividing both sides by 5 to solve for x:

x = 18

Now we substitute x back into 2x, which represents the measure of the smaller angle:

2(18) = 36

So, the measure of the smaller angle is 36°. The correct answer is A) 36°.

Using the Law of Sines in triangle PQR, we have:

sin(P) / PQ = sin(Q) / QR

Given the angle P is 30 degrees, angle Q is 60 degrees, and side PQ has a length of 10, we can substitute the values into the formula:

sin(30) / 10 = sin(60) / QR

Simplifying:

1/2 / 10 = √3/2 / QR

Now, we can cross-multiply to solve for QR:

QR = (10 * √3/2) / (1/2)
QR = 10√3

So, the length of side QR is 10√3. The correct answer is B) 10√3.

To find the coordinates of the midpoint (M) of a line segment with endpoints A(x₁, y₁) and B(x₂, y₂), use the midpoint formula:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Given the points (8, 10) and (4, 2), substitute the coordinates into the formula:

M = ((8 + 4) / 2, (10 + 2) / 2)
M = (12 / 2, 12 / 2)
M = (6, 6)

So, the coordinates of the midpoint are (6, 6). The correct answer is C) (6, 6).

The sum of the interior angles of a polygon can be found using the formula:

sum = (n - 2) × 180°

where n is the number of sides of the polygon.

For a hexagon, the number of sides is 6:

sum = (6 - 2) × 180°
sum = 4 × 180°
sum = 720°

So, the sum of the interior angles of a hexagon is 720°. The correct answer is D) 720°.