5.4 use medians and altitudes

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