8. when do the angle bisectors of a triangle intersect? a. circumcenter b. incenter c. orthocenter d. centroid 9. what must be shown to confirm that a quadrilateral is a rectangle using coordinates? a. opposite sides have the same length b. all sides are of equal length c. opposite sides are parallel and all angles are right angles d. diagonals intersect at right angles 10. a square is inscribed in a circle with radius r. what is the area of the square? a. r² b. 2r² c. 2r³ d. r²√2 11. what is the length of a line - segment between points (2, - 2) and (-5, 6)? a. 4 units b. 12 units c. 6.3 units d. 10.4 units 12. in aerospace engineering, what is the primary reason honeycombs are used in honeycomb structures? a. to maximize strength while minimizing weight b. to increase weight c. to reduce manufacturing time d. all of the above 13. given a parallelogram with vertices at (1, 1), (5, 1), (8, 4), and (4, 4), which condition proves that the diagonals bisect each other? a. diagonal lengths are equal b. slopes of the diagonals are equal c. slopes of opposite sides are equal d. midpoints of diagonals are equal 14. are two lines perpendicular if their slopes are ⅓ and - 3? a. yes b. the answer is indeterminate c. a possible in this situation

Question

Accepted Answer

### 8.
# Brief Explanations:
The in - center of a triangle is the point of intersection of the angle bisectors of a triangle. The ortho - center is the intersection of the altitudes, the centroid is the intersection of the medians.
# Answer:
b. Incenter

### 9.
# Brief Explanations:
A rectangle is a parallelogram with all angles being right - angles. Opposite sides being parallel is a property of parallelograms in general, and for a rectangle, all angles must be right angles in addition to opposite sides being parallel. Equal side lengths are for a square, and diagonals intersecting at right angles is for a rhombus or square.
# Answer:
c. Opposite sides are parallel and all angles are right angles

### 10.
# Explanation:
## Step1: Recall the relationship between the square and the circle.
The diameter of the circle is the diagonal of the square. If the radius of the circle is $r$, the diameter $d = 2r$. Let the side length of the square be $a$. Using the Pythagorean theorem for the square ($d^{2}=a^{2}+a^{2}$, since the diagonal of a square with side length $a$ forms a right - triangle with two sides of the square), and $d = 2r$, so $(2r)^{2}=2a^{2}$.
## Step2: Solve for $a^{2}$ (the area of the square).
$4r^{2}=2a^{2}$, then $a^{2}=2r^{2}$. The area of a square $A=a^{2}$, so the area of the square is $2r^{2}$.
# Answer:
c. $2r^{2}$

### 11.
# Explanation:
## Step1: Use the distance formula.
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$. Here, $x_1 = 2,y_1=-2,x_2=-5,y_2 = 6$.
## Step2: Substitute the values.
$d=\sqrt{(-5 - 2)^{2}+(6+2)^{2}}=\sqrt{(-7)^{2}+8^{2}}=\sqrt{49 + 64}=\sqrt{113}\approx10.63\approx10.6$ (closest to 10.6 which is not in the options, but the closest value among the options is 10.4 which may be due to approximation differences in the problem - setting). The correct calculation gives $d=\sqrt{( - 5-2)^{2}+(6 + 2)^{2}}=\sqrt{(-7)^{2}+8^{2}}=\sqrt{49+64}=\sqrt{113}\approx10.63$. If we assume some rounding in the problem, we note that $\sqrt{113}$ is closest to 10.4 among the given options.
# Answer:
d. 10.4 units

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

8.

Step1: Recall the relationship between the square and the circle.

Step2: Solve for \(a^{2}\) (the area of the square).

Answer:

9.