Question

1. where do the angle bisectors of a triangle intersect?

a. circumcenter
b. incenter
c. orthocenter
d. centroid

1. 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

1. a square is inscribed in a circle with radius r. what is the area of the square?

a. r²
b. 2r²
c. 3r²
d. r²√2

1. what is the length of a line - segment between points (2, - 2) and (- 5, 6)?

a. 4 units
b. 11 units
c. 8.5 units
d. 10.6 units

Explanation:

• For the first question, the in - center of a triangle is the point of intersection of the angle bisectors. The circum - center is the intersection of the perpendicular bisectors of the sides, the ortho - center is the intersection of the altitudes, and the centroid is the intersection of the medians.
• For the second question, a rectangle is defined as a quadrilateral with opposite sides parallel and all angles being right angles. Just having opposite sides of the same length makes it a parallelogram, all sides of equal length makes it a rhombus, and diagonals intersecting at right angles makes it a rhombus.
• For the third question, if a square is inscribed in a circle of radius \(r\), the diameter of the circle is the diagonal of the square. Let the side of the square be \(a\). Using the Pythagorean theorem, if the diagonal \(d = 2r\), then \(d^{2}=a^{2}+a^{2}\), so \((2r)^{2}=2a^{2}\), and \(a^{2}=2r^{2}\), which is the area of the square.
• For the fourth question, the distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\). Here \(x_1 = 2,y_1=-2,x_2=-5,y_2 = 6\). Then \(d=\sqrt{(-5 - 2)^{2}+(6+2)^{2}}=\sqrt{(-7)^{2}+8^{2}}=\sqrt{49 + 64}=\sqrt{113}\approx10.6\) units.

Answer:

1. b. Incenter
2. c. Opposite sides are parallel and all angles are right angles
3. b. \(2r^{2}\)
4. d. 10.6 units