Centroid Day 2: Notes Packet

Name:______Date:______Period:______

Centroid Day 2: Notes Packet

Do Now: Complete the chart

Center of the Triangle / At the concurrency of / Located(inside, outside, on, etc…) / Create a circle?
Orthocenter / Acute:
Right:
Obtuse:
Centroid / Always:
Incenter / Always:
Circumcenter / Acute:
Right:
Obtuse:

All Of My Children Are Bringing In 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:

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.

1.) In , medians and are concurrent at point Q. If TQ = 3x – 1 and TM = x + 1, what is the length of median ? Justify your answer.

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

(4) 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.

1.) In the diagram below, is a median of triangle PQR and point C is the centroid of triangle PQR. If QC = 5x and CM = x + 12, determine and state the length of .

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 and are concurrent at point Q. If TQ = 9x–3, and QM = 3x+3, what is the length of median .

4.) In shown below, P is the centroid and BF = 18.

What is the length of ?

(1) 6(2) 9(3) 3(4) 12

5.) In triangle ACE, is the median to , is the median to and is the median to . The medians meet at Q. If AQ = 48, find the measure of .