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