GT Geom. Drill 12/13/13
Define each of the following.
1. A perpendicular bisector.
2. An angle bisector.

3. Find the midp...
Objectives
Prove and apply theorems about
perpendicular bisectors.
Prove and apply theorems about angle
bisectors.
Vocabulary
equidistant
locus
When a point is the same
distance from two or more
objects, the point is said to be
equidistant from the objects.
Triangle...
Based on these theorems, an angle bisector can be
defined as the locus of all points in the interior of the
angle that are...
A locus is a set of points that
satisfies a given condition. The
perpendicular bisector of a
segment can be defined as the...
Example 1A: Applying the Perpendicular Bisector
Theorem and Its Converse
Find each measure.
MN
MN = LN

 Bisector Thm.

M...
Example 1B: Applying the Perpendicular Bisector
Theorem and Its Converse
Find each measure.
BC
Since AB = AC and
, is the
...
Example 1C: Applying the Perpendicular Bisector
Theorem and Its Converse

Find each measure.

TU
TU = UV

 Bisector Thm.
...
Remember that the distance between a point and a
line is the length of the perpendicular segment from
the point to the lin...
Example 2A: Applying the Angle Bisector Theorem
Find the measure.

BC
BC = DC

 Bisector Thm.

BC = 7.2

Substitute 7.2 f...
Example 2B: Applying the Angle Bisector Theorem
Find the measure.

mEFH, given that mEFG = 50°.
Since EH = GH,
and
,
bis...
Example 2C: Applying the Angle Bisector Theorem

Find mMKL.
Since, JM = LM,

and

,
bisects JKL
by the Converse of the A...
Example 4: Writing Equations of Bisectors in the
Coordinate Plane
Write an equation in point-slope form for the
perpendicu...
Example 4 Continued
Step 2 Find the midpoint of

.

Midpoint formula.

mdpt. of

=
Example 4 Continued
Step 3 Find the slope of the perpendicular bisector.
Slope formula.

Since the slopes of perpendicular...
Example 4 Continued
Step 4 Use point-slope form to write an equation.
The perpendicular bisector of

has slope

and passes...
Example 4 Continued
Check It Out! Example 4
Write an equation in point-slope form for the
perpendicular bisector of the segment with
endpoints...
Check It Out! Example 4 Continued
Step 2 Find the midpoint of PQ.
Midpoint formula.
Check It Out! Example 4 Continued
Step 3 Find the slope of the perpendicular bisector.
Slope formula.

Since the slopes of...
Check It Out! Example 4 Continued
Step 4 Use point-slope form to write an equation.
The perpendicular bisector of PQ has s...
Use the diagram for Items 1–2.
1. Given that mABD = 16°, find mABC. 32°
2. Given that mABD = (2x + 12)° and mCBD =
(6x...
Assignment #48?

• Handout from book
• P304-305 (myhrw.com)#12- 32, for 30 and
31 two column proof
• Here is the picture b...
Chapter 5 unit f 001
Chapter 5 unit f 001
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Chapter 5 unit f 001

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Chapter 5 unit f 001

  1. 1. GT Geom. Drill 12/13/13 Define each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and slope of the segment (2, 8) and (–4, 6).
  2. 2. Objectives Prove and apply theorems about perpendicular bisectors. Prove and apply theorems about angle bisectors.
  3. 3. Vocabulary equidistant locus
  4. 4. When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points.
  5. 5. Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle.
  6. 6. A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment.
  7. 7. Example 1A: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. MN MN = LN  Bisector Thm. MN = 2.6 Substitute 2.6 for LN.
  8. 8. Example 1B: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. BC Since AB = AC and , is the perpendicular bisector of by the Converse of the Perpendicular Bisector Theorem. BC = 2CD Def. of seg. bisector. BC = 2(12) = 24 Substitute 12 for CD.
  9. 9. Example 1C: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. TU TU = UV  Bisector Thm. 3x + 9 = 7x – 17 Substitute the given values. 9 = 4x – 17 Subtract 3x from both sides. 26 = 4x 6.5 = x Add 17 to both sides. Divide both sides by 4. So TU = 3(6.5) + 9 = 28.5.
  10. 10. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line.
  11. 11. Example 2A: Applying the Angle Bisector Theorem Find the measure. BC BC = DC  Bisector Thm. BC = 7.2 Substitute 7.2 for DC.
  12. 12. Example 2B: Applying the Angle Bisector Theorem Find the measure. mEFH, given that mEFG = 50°. Since EH = GH, and , bisects EFG by the Converse of the Angle Bisector Theorem. Def. of  bisector Substitute 50° for mEFG.
  13. 13. Example 2C: Applying the Angle Bisector Theorem Find mMKL. Since, JM = LM, and , bisects JKL by the Converse of the Angle Bisector Theorem. mMKL = mJKM Def. of  bisector 3a + 20 = 2a + 26 a + 20 = 26 a=6 Substitute the given values. Subtract 2a from both sides. Subtract 20 from both sides. So mMKL = [2(6) + 26]° = 38°
  14. 14. Example 4: Writing Equations of Bisectors in the Coordinate Plane Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1). Step 1 Graph . The perpendicular bisector of is perpendicular to at its midpoint.
  15. 15. Example 4 Continued Step 2 Find the midpoint of . Midpoint formula. mdpt. of =
  16. 16. Example 4 Continued Step 3 Find the slope of the perpendicular bisector. Slope formula. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is
  17. 17. Example 4 Continued Step 4 Use point-slope form to write an equation. The perpendicular bisector of has slope and passes through (8, –2). y – y1 = m(x – x1) Point-slope form Substitute –2 for y1, for x1. for m, and 8
  18. 18. Example 4 Continued
  19. 19. Check It Out! Example 4 Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P(5, 2) and Q(1, –4). Step 1 Graph PQ. The perpendicular bisector of is perpendicular to at its midpoint.
  20. 20. Check It Out! Example 4 Continued Step 2 Find the midpoint of PQ. Midpoint formula.
  21. 21. Check It Out! Example 4 Continued Step 3 Find the slope of the perpendicular bisector. Slope formula. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is .
  22. 22. Check It Out! Example 4 Continued Step 4 Use point-slope form to write an equation. The perpendicular bisector of PQ has slope passes through (3, –1). y – y1 = m(x – x1) Point-slope form Substitute. and
  23. 23. Use the diagram for Items 1–2. 1. Given that mABD = 16°, find mABC. 32° 2. Given that mABD = (2x + 12)° and mCBD = (6x – 18)°, find mABC. 54° Use the diagram for Items 3–4. 3. Given that FH is the perpendicular bisector of EG, EF = 4y – 3, and FG = 6y – 37, find FG. 65 4. Given that EF = 10.6, EH = 4.3, and FG = 8.6 10.6, find EG.
  24. 24. Assignment #48? • Handout from book • P304-305 (myhrw.com)#12- 32, for 30 and 31 two column proof • Here is the picture blown up for 23-28

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