1.
In a triangle, a perpendicular bisector is __________________ to a side of the triangle and intersects that side at its ____________________. Ckeck the choices that best fit the blanks.
Correct Answer(s)
A. Perpendicular
C. Midpoint
Explanation
A perpendicular bisector is a line that is perpendicular to a side of the triangle and intersects that side at its midpoint. This means that the line cuts the side into two equal segments, with the point of intersection being exactly halfway between the endpoints of the side.
2.
True or False. In a triangle, an angle bisector divides an angle into two congruent angles.
Correct Answer
A. True
Explanation
An angle bisector is a line or ray that divides an angle into two equal angles. In a triangle, an angle bisector divides one of the angles into two congruent angles, meaning that the two resulting angles have the same measure. Therefore, the statement "an angle bisector divides an angle into two congruent angles" is true.
3.
True or False. In a triangle, a median connects a vertex with the midpoint of the opposite side.
Correct Answer
A. True
Explanation
Look at the pictures of medians, where is the median segment intersecting the triangle.
4.
In a triangle, an altitude connects a midpoint with the line containing the opposite side, and is perpendicular to that line.
Correct Answer
B. False
Explanation
An altitude always uses a vertex.
5.
Choose the best appropriate answer for the blank. In an _____________ triangle, the circumcenter, incenter, centroid, and orthocenter are the same point.
Correct Answer
C. Equilateral
Explanation
In an equilateral triangle, all three sides and angles are equal. This symmetry allows the circumcenter (the point where the perpendicular bisectors of the sides intersect), the incenter (the point where the angle bisectors intersect), the centroid (the point where the medians intersect), and the orthocenter (the point where the altitudes intersect) to all coincide at the same point. This is because the equilateral triangle has a high degree of symmetry, resulting in all these special points overlapping.
6.
Choose the best appropriate answer for the black. In an isosceles triangle, the altitude from the _________________ to the base is also a perpendicular bisector, an angle bisector, and a median.
Correct Answer
B. Vertex
Explanation
In an isosceles triangle, the altitude from the vertex to the base is also a perpendicular bisector, an angle bisector, and a median. This is because the altitude is a line segment that is perpendicular to the base and passes through the vertex. As a perpendicular bisector, it divides the base into two equal segments. As an angle bisector, it divides the vertex angle into two equal angles. And as a median, it connects the vertex to the midpoint of the base, dividing the triangle into two congruent triangles. Therefore, the altitude from the vertex to the base in an isosceles triangle has multiple properties.
7.
Always, Sometimes or Never. A median of a triangle _____________________ has a midpoint as an endpoint.
Correct Answer
A. Always
Explanation
A median of a triangle always has a midpoint as an endpoint. This is because a median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. Since the midpoint of a line segment is always equidistant from the endpoints, the midpoint of the opposite side will always lie on the median. Therefore, the statement is always true.
8.
Always, Sometimes, or Never. A median of a triangle ______________________ lies outside of the triangle.
Correct Answer
C. Never
Explanation
A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. Since the midpoint of a side always lies within the triangle, and the vertex obviously lies on the triangle, the median will always lie within the triangle. Therefore, the correct answer is "Never".
9.
Always, Sometimes, or Never. A perpendicular bisector of a triangle _____________ contains a vertex of the triangle.
Correct Answer
B. Sometimes
Explanation
A perpendicular bisector of a triangle sometimes contains a vertex of the triangle because it depends on the specific triangle. In some cases, the perpendicular bisector will intersect with a vertex of the triangle, while in other cases it will not. Therefore, it cannot be said that it always or never contains a vertex of the triangle.
10.
Always, Sometimes, or Never. The angle bisectors of a triangle _________________ intersect at a single point.
Correct Answer
A. Always
Explanation
The angle bisectors of a triangle always intersect at a single point. This point is called the incenter of the triangle. The incenter is equidistant from the three sides of the triangle, and it is the center of the inscribed circle in the triangle. Therefore, it is always true that the angle bisectors of a triangle intersect at a single point.
11.
Always, Sometimes, or Never. The circumcenter of a triangle __________________ lies outside the triangle.
Correct Answer
B. Sometimes
Explanation
The circumcenter of a triangle sometimes lies outside the triangle because the location of the circumcenter depends on the type of triangle. In an acute triangle, the circumcenter will always lie inside the triangle. In a right triangle, the circumcenter will always lie on the midpoint of the hypotenuse. However, in an obtuse triangle, the circumcenter will always lie outside the triangle. Therefore, the statement "The circumcenter of a triangle sometimes lies outside the triangle" is correct.
12.
Always, Sometimes, or Never. The centroid of a triangle _____________________ lies outside the triangle.
Correct Answer
C. Never
Explanation
The centroid of a triangle is the point of intersection of the medians of the triangle. The medians always intersect within the triangle, which means that the centroid always lies within the triangle. Therefore, the statement "The centroid of a triangle never lies outside the triangle" is correct.
13.
Correct Answer
B. (7, -4)
14.
Correct Answer
B. (-3, -4)
15.
Which of the following equations best matches the line containing the altitude of triangle ABC through vertex B?
Correct Answer
B. Y = (5/3)x + 2
Explanation
The equation y = (5/3)x + 2 represents a line with a slope of 5/3 and a y-intercept of 2. This equation matches the line containing the altitude of triangle ABC through vertex B because it has the same slope and intercept as the line.
16.
Which of the following equations best matches the line containing the median of triangle ABC through vertex A?
Correct Answer
C. Y = -x-2
Explanation
The equation y = -x-2 is the best match for the line containing the median of triangle ABC through vertex A because it has the same slope (-1) as the median and passes through the point A (-2, 0), which is the vertex A of the triangle.
17.
V is the centroid of triangle PRT shown below. If QV = 3, then QT = _____ .
Correct Answer
A. 9
Explanation
Since V is the centroid of triangle PRT, it divides each median into two segments such that the segment joining V to the vertex is twice as long as the segment joining V to the midpoint of the opposite side. Therefore, if QV = 3, then QT would be equal to 3 * 2 = 6.
18.
V is the centroid of triangle PRT shown below. If VS = 7, then PV = _____.
Correct Answer
B. 14
Explanation
Since V is the centroid of triangle PRT, it divides the median PS into two segments, with PV being one of them. The segment PV is twice as long as the segment VS. Therefore, if VS = 7, then PV would be twice that length, which is 14.
19.
V is the centroid of triangle PRT shown below. QV = 3m – 1 and QT = 7m + 3.
Find the value of m, then find VT.
Correct Answer
B. M=3, VT = 16
Explanation
Since V is the centroid of triangle PRT, it divides the medians in a 2:1 ratio. Given that QV = 3m - 1 and QT = 7m + 3, we can set up the equation 3m - 1 = 2(7m + 3). Solving for m, we get m = 3. To find VT, we substitute m = 3 into QT and solve for VT. VT = 7(3) + 3 = 21 + 3 = 24. Therefore, the correct answer is m = 3 and VT = 24.
20.
Using the figure below, Triangle ABC is isosceles and <A is the vertex angle. AD is the perpendicular bisector of BC. If BC = 2x^{2} and BD = 4x + 21 find the value(s) of x.
Correct Answer
C. X = 7 or -3
Explanation
In an isosceles triangle, the perpendicular bisector of the base divides it into two congruent segments. In this case, AD is the perpendicular bisector of BC, so BD and CD are congruent. Given that BD = 4x + 21, we can set it equal to CD and solve for x. Therefore, 4x + 21 = 2x^2. Rearranging the equation, we get 2x^2 - 4x - 21 = 0. Factoring this quadratic equation, we find (2x + 7)(x - 3) = 0. Setting each factor equal to zero, we get x = -7 or x = 3. Therefore, the value(s) of x are 7 or -3.