5.45.4 Use Medians and AltitudesBell Thinger
1. For A(–4, 8) and B(5, 8), find the midpoint of AB.
2. For A(–3, 2) and B(4, –1), find the length of AB.
ANSWER , 821
ANSWER 58
ANSWER 12
3. For A(0, 4) and C(18, 4), find the length of AB, where B is a point the distance from A to C.
32
5.4
5.4
5.4Example 1
SOLUTION
SQ = 23
SW Concurrency of Medians of a Triangle Theorem
8 = 23
SW Substitute 8 for SQ.
12 = SW Multiply each side by the reciprocal, .32
Then QW = SW – SQ = 12 – 8 = 4.
So, QW = 4 and SW = 12.
In RST, Q is the centroid and SQ = 8. Find QW and SW.
5.4Example 2
SOLUTION
Sketch FGH. Then use the Midpoint Formula to find the midpoint K of FH and sketch median GK .
The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.
K( ) =2 + 6 , 5 + 12 2 K(4, 3)
5.4
The distance from vertex G(4, 9) to K(4, 3) is
9 – 3 = 6 units. So, the centroid is (6) = 4 units
down from G on GK .
23
The coordinates of the centroid P are (4, 9 – 4), or (4, 5).
The correct answer is B.
Example 2
5.4Guided Practice
There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P.
1. If SC = 2100 feet, find PS and PC.
700 ft, 1400 ftANSWER
2. If BT = 1000 feet, find TC and BC.
1000 ft, 2000 ftANSWER3. If PT = 800 feet, find PA and TA.
1600 ft, 2400 ftANSWER
P
5.4
5.4
5.4Example 3
Find the orthocenter P in an acute, a right, and an obtuse triangle.
SOLUTION
Acute triangle
P is inside triangle.
Right triangle
P is on triangle.
Obtuse triangle
P is outside triangle.
5.4Guided Practice
6. Triangle PQR is an isosceles triangle and OQ is an altitude. What else do you know about OQ ? What are the coordinates of P?
OQ is also a perpendicular bisector, angle bisector, and median; (–h, 0).
ANSWER
5.4Exit Slip
In Exercises 1–3, use the diagram.G is the centroid of ∆ABC.
1. If BG = 9, find BF.
ANSWER 13.5
2. If BD = 12, find AD.
ANSWER 12
3. If CD = 27, find GC.
ANSWER 18
5.4Exit Slip
ANSWER (1, 1)
5. Which type of triangle has its orthocenter on the triangle?
ANSWER a right triangle
4. Find the centroid of ∆ABC.
5.4
Homework
Pg 336-339#6, 9, 16, 34, 35