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Centroid and Orthocenter Answers

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1
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A centroid is the intersection of three

A
altitudes in a triangle.
B
perpendicular bisectors in a triangle.
C
angle bisectors in a triangle.
D
medians in a triangle.
2
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If an orthocenter lies inside of a triangle, then the triangle must be

A
isosceles.
B
obtuse.
C
right.
D
acute.
3

Point G is the centroid of triangle ABC. AG = (5x + 4) units and GF = (3x – 1) units.

Question illustration
A
11 units
B
15 units
C
43 units
D
51 units
4

Triangle H J K is shown. Lines are drawn from each point to the opposite side and intersect at point G to form line segments H C, J E, and K D.

Question illustration
A
Point G cannot be a centroid because JG does not equal GE.
B
Point G cannot be a centroid because JG and GE are in the ratio 1:2.
C
Point G can be a centroid because GE and JG are in the ratio 2:1.
D
Point G can be a centroid because JG + GE = JE.
5

In triangle TRS, TZ = (3x) inches and WZ = (2x - 3) inches.[Figure may not be drawn to scale]

Question illustration
A
6 inches
B
9 inches
C
18 inches
D
27 inches
6

The steps shown can be used to prove that the medians of a triangle meet at a point.1. Define segments BD and CE as medians of triangle ABC.2. Write linear equations for and .3. Use a system of linear equations to solve for the coordinates of intersection point G.4. Write the equation of .5. Write an expression for the midpoint of BC, point F. 6. Show that point F lies on .7. ?

Question illustration
A
Write a linear equation for each side of the triangle.
B
Write an expression for the midpoint of AC and BC.
C
Show that AF is congruent to BD and CE.
D
Show that AF is the median of BC.
7

In triangle NLM, point S is the centroid, QS = (3x – 5) cm, and NS = (4x) cm.

Question illustration
A
5 cm
B
10 cm
C
20 cm
D
30 cm
8

Triangle A B C is shown. Lines are drawn from each point to the opposite side and intersect at point D.

Question illustration
A
Point D can be the orthocenter because it is the point of intersection of three segments coming from the vertices of the triangle.
B
Point D can be the orthocenter because each vertex angle appears to be bisected.
C
Point D cannot be the orthocenter because the orthocenter of an obtuse triangle is located outside the triangle.
D
Point D cannot be the orthocenter because the orthocenter of an obtuse triangle is located on the perimeter of the triangle.
9

In which figure is point G an orthocenter?

A
Triangle A B C is a right triangle. Lines are drawn from each point to the opposite side and intersect at point G.
Option A
B
Triangle F D E is shown. Lines are drawn from each point to the opposite side and intersect at point G. The lines cut each side into 2 equal parts.
Option B
C
Triangle L M N is shown. Lines are drawn from each point to the opposite side and intersect at point G. Each angle has a different measure.
Option C
D
Triangle H J K is shown. Lines are drawn from each point to the opposite side to form right angles and the lines intersect at point G.
Option D
10

Point G is the centroid of triangle ABC. The length of segment CG is 6 units greater than the length of segment DG.

Question illustration
A
6 units
B
12 units
C
18 units
D
24 units
11

Triangle D E F is shown. Lines are drawn from each point to the opposite side and intersect at point G. Line segments D C, E B, and F A are formed and cut each side into 2 equal parts.

Question illustration
A
5 cm
B
10 cm
C
15 cm
D
20 cm
12

In triangle ABC, EG = x inches and BG = (5x - 12) inches. [Figure may not be drawn to scale]

Question illustration
A
2 inches
B
4 inches
C
8 inches
D
12 inches
13

In which type of triangle is the orthocenter on the perimeter of the triangle?

A
a right triangle
B
an acute triangle
C
an obtuse triangle
D
an equilateral triangle
14

Point G is the centroid of triangle ABC. The length of segment CG is 6 units greater than the length of segment DG.

Question illustration
A
6 units
B
12 units
C
18 units
D
24 units
15

In the diagram, GB = 2x + 3..

Question illustration
A
5 units
B
10 units
C
15 units
D
30 units
16

An orthocenter is the intersection of three

A
angle bisectors in a triangle.
B
altitudes in a triangle.
C
medians in a triangle.
D
perpendicular bisectors in a triangle.
17

Triangle D E F is shown. Lines are drawn from each point to the opposite side and intersect at point G. Line segments D C, E B, and F A are formed and cut each side into 2 equal parts.

Question illustration
A
5 cm
B
10 cm
C
15 cm
D
20 cm
18

In triangle DEF, CG = (x + 5) units and DG = (3x - 2) units.[Figure may not be drawn to sacle]

Question illustration
A
12 units
B
17 units
C
34 units
D
51 units
19

The steps shown can be used to prove that the medians of a triangle meet at a point.1. Define segments BD and CE as medians of triangle ABC.2. Write linear equations for and .3. Use a system of linear equations to solve for the coordinates of intersection point G.4. Write the equation of .5. Write an expression for the midpoint of BC, point F. 6. Show that point F lies on .7. ?

Question illustration
A
Write a linear equation for each side of the triangle.
B
Write an expression for the midpoint of AC and BC.
C
Show that AF is congruent to BD and CE.
D
Show that AF is the median of BC.
20

Triangle H J K is shown. Lines are drawn from each point to the opposite side and intersect at point G to form line segments H C, J E, and K D.

Question illustration
A
Point G cannot be a centroid because JG does not equal GE.
B
Point G cannot be a centroid because JG and GE are in the ratio 1:2.
C
Point G can be a centroid because GE and JG are in the ratio 2:1.
D
Point G can be a centroid because JG + GE = JE.

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