4.6 PreAP Special Segments

1. True or False. Find the point of concurrency of the perpendicular bisectors. The point of concurrency is on the hypotenuse of the triangle.

True

False

The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle. It is the center of the circle that passes through all three vertices of the triangle. Since the hypotenuse is one of the sides of the triangle, the circumcenter must lie on it. Therefore, the statement is true.

Explanation

The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle. It is the center of the circle that passes through all three vertices of the triangle. Since the hypotenuse is one of the sides of the triangle, the circumcenter must lie on it. Therefore, the statement is true.

2. True or False. Find the point of concurrency of the altitudes. The point of concurrency is inside the triangle.

True

False

The point of concurrency of the altitudes is at the 90 degree angle.

Explanation

The point of concurrency of the altitudes is at the 90 degree angle.

3. 2. Which of the following BEST describes the special segment UT?

Median

Altitude

Perpendicular Bisector

Angle Bisector

The special segment UT is an angle bisector. An angle bisector is a line or ray that divides an angle into two congruent angles. In this case, the segment UT divides an angle into two equal angles, making it an angle bisector.

Explanation

The special segment UT is an angle bisector. An angle bisector is a line or ray that divides an angle into two congruent angles. In this case, the segment UT divides an angle into two equal angles, making it an angle bisector.

4. Which of the following BEST describes the special segment UT?

Median

Altitude

Perpendicular Bisector

Angle Bisector

An altitude is a line segment drawn from a vertex of a triangle perpendicular to the opposite side. It is used to find the height or distance between the vertex and the opposite side. In this case, the special segment UT is described as an altitude, indicating that it is a line segment drawn from vertex U perpendicular to the opposite side T.

Explanation

An altitude is a line segment drawn from a vertex of a triangle perpendicular to the opposite side. It is used to find the height or distance between the vertex and the opposite side. In this case, the special segment UT is described as an altitude, indicating that it is a line segment drawn from vertex U perpendicular to the opposite side T.

5. Which of the following BEST describes the special segment UT?

Median

Altitude

Perpendicular Bisector

Angle Bisector

A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. It divides the triangle into two equal areas. In the context of the question, the special segment UT is described as a median because it connects a vertex U to the midpoint T of the opposite side.

Explanation

A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. It divides the triangle into two equal areas. In the context of the question, the special segment UT is described as a median because it connects a vertex U to the midpoint T of the opposite side.

6. You will need graph paper for this problem. Graph triangle CAR using the points C(-4, 0), A(2, -4), and R(3, 4). Then answer the questions.

Midpoint of segment CA.

Slope of segment CA.

Slope of segment RS.

CA and RS are_______ lines

RS is a/an

Submit

7. Draw the altitude from point G. What is the slope of the altitude?

8/4 or 2/1

-4/8 or -1/2

7/3

6/8 or 3/4

The slope of the altitude from point G can be determined by finding the slope of the line passing through point G and the orthocenter of the triangle. Since the orthocenter is the intersection of the altitudes, the slope of the altitude is the negative reciprocal of the slope of the line passing through the orthocenter and the midpoint of the opposite side. In this case, the slope is -1/2 or -4/8.

Explanation

The slope of the altitude from point G can be determined by finding the slope of the line passing through point G and the orthocenter of the triangle. Since the orthocenter is the intersection of the altitudes, the slope of the altitude is the negative reciprocal of the slope of the line passing through the orthocenter and the midpoint of the opposite side. In this case, the slope is -1/2 or -4/8.

8. Draw the altitude from point Q. What is the slope of the altitude?

6/9 or 2/3

-6/9 or -2/3

9/6 or 3/2

-9/6 or -3/2

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Explanation

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9. Graph triangle ABC using points A(4, -2), B(-2, 5) and C(0, -4). Find the midpoint of segment AB. Label this point D on the graph. Connect D to C. Match the choices below with the correct answer.

Special Segment CD

midpoint of AB

extra answer

Submit

10. Graph triangle ABC using points A(4, -2), B(-2, 5) and C(0, -4). Find the midpoint of CA. Label this point E on the graph. Connect point E to point B. Now find the slope of BE and AC. Match the following statements.