Name:_________________________________________________
Date:________ Period:_______
Centroid Day 2: Notes Packet
Do Now: Complete the chart
Center of the Triangle
Orthocenter
At the concurrency of
Centroid
Located (inside, outside, on, etc…)
Acute:
Right:
Obtuse:
Always:
Incenter
Always:
Circumcenter
Acute:
Right:
Obtuse:
All Of
My Children
Are Bringing In
Create a circle?
Peanut Butter Cookies
Concurrency of the Medians:
Recall: The ___________________ of a triangle is the line segment that joins the vertex to the
______________ of the opposite side of the triangle.
The three medians of a triangle are concurrent in a point that is called the ___________________.
There is a special relationship that involves the line segments when all of the three medians meet.
The distance from each vertex to the centroid is ________________________ of the length of the entire
median drawn from that vertex.
Let’s Take a Look at the Diagram:
AO = __________
CO = _________
BO = ________
In addition, the distance from each centroid to the opposite side (midpoint) is one-third of the distance
of the entire median.
OF= __________
OE = _________
OD = ________
The centroid also divides the median into two segments in the ratio 2:1, such that:
𝐴𝑂
𝑂𝐸
=
𝐶𝑂
𝑂𝐷
=
𝐵𝑂
𝑂𝐹
=
If you notice, the bigger part of the ratio is the segment that is drawn from the vertex to the centroid.
The smaller part of the median is always the part that is drawn from the centroid to the midpoint of the
opposite side.
When working with these ratios, it is important to never mix the two up!
It can sometimes be helpful to label the diagram with the ratios. Since the vertex to the centroid can be
labeled as ______ and from the centroid to the midpoint of the opposite side can be labeled as ______,
the entire median is known as _______.
Therefore, whenever we do these centroid problems, we always set up a proportion.
̅̅̅̅ are concurrent at point Q. If TQ = 3x – 1 and TM = x + 1, what is the
̅̅̅̅̅ and 𝑆𝑃
1.) In ∆𝑅𝑆𝑇, medians 𝑇𝑀
̅̅̅̅̅? Justify your answer.
length of median 𝑇𝑀
2.) In ΔABC, points J, K, and L are midpoints of sides AB, BC, and AC, respectively. If the three medians of
the triangle intersect at point P and the length of LP is 6, what is the length of BL?
3.) In triangle ABC, medians AD, BE, and CF are concurrent at point P. If AD = 24 inches, find the length of
AP.
4.) In the diagram Jose found centroid P by constructing the three medians. He measured CF and found
it to be 6 inches. If PF = x, then what equation can be used to find the value of x?
(1) x + x = 6
(2) 2x + x = 6
(3) 3x + 2x = 6
2
(4) x + 3 𝑥 = 6
5.) In ΔABC, medians CF and AD intersect at P. If CF = 6x and CP = 5x – 6, find x, CF, CP, and PF.
Name:_________________________________________________
Date:________ Period:_______
Centroid: Day 2 Homework
Complete the following examples. Show all work, including formulas. Make sure to justify all solutions.
̅̅̅̅̅ is a median of triangle PQR and point C is the centroid of triangle PQR. If
1.) In the diagram below, 𝑄𝑀
̅̅̅̅̅.
QC = 5x and CM = x + 12, determine and state the length of 𝑄𝑀
2.) The three medians of a triangle intersect at a point. Which measurements could represent the
segments of one of the medians?
(1) 2 and 3
(2) 3 and 4.5
(3) 3 and 6
(4) 3 and 9
3.) In ΔRST, medians ̅̅̅̅̅
𝑇𝑀 and ̅̅̅̅
𝑆𝑃 are concurrent at point Q. If TQ = 9x–3, and QM = 3x+3, what is the
̅̅̅̅̅
length of median 𝑇𝑀.
4.) In ∆𝐴𝐵𝐶 shown below, P is the centroid and BF = 18.
What is the length of ̅̅̅̅
𝐵𝑃?
(1) 6
(2) 9
(3) 3
(4) 12
5.) In triangle ACE, ̅̅̅̅
𝐸𝐷 is the median to ̅̅̅̅
𝐴𝐶 , ̅̅̅̅
𝐶𝐹 is the median to ̅̅̅̅
𝐴𝐸 and ̅̅̅̅
𝐴𝐻 is the median to ̅̅̅̅
𝐶𝐸 . The
̅̅̅̅.
medians meet at Q. If AQ = 48, find the measure of 𝐴𝐻