Ch 6 – More About Triangles
6.1 – Medians
Median:
Hands-On Geometry: P228
Complete Steps
Do “Try These” on same triangle
Answer: What do you notice about the medians of a triangle?
Example: In ABC , CE and AD are medians.
Find BE if AB = 18.
If CD = 2x + 5, BD = 4x – 1, and AE = 5x – 2, find BE.
Example: In MNP , MC and ND are medians.
What is NC if NP = 18?
If DP = 7.5, find MP.
Example: In RST , RP and SQ are medians. If RQ = 7x – 1, SP = 5x – 4, and
QT = 6x + 9, find PT.
Centroid:
Concurrent:
Theorem 6.1:
Example: In XYZ , XP and ZN are medians.
Find ZQ if QN = 5.
If XP = 1-.5, what is QP?
Example: In ABC , CD , BF , and AE are medians.
If CG = 14, what is DG?
Find the measure of BF if GF = 6.8.
Example: In XYZ , XP , ZN , and YM are medians.
Find YQ if QM = 4.
If QZ = 18, what is ZN?
6.2 – Altitudes and Perpendicular Bisectors
Altitude:
Example: Tell whether each segment is an altitude of the triangle.
Perpendicular Bisector:
Example: Tell whether each line or segment is a perpendicular bisector of a side of the triangle.
Special Occasions:
Example: Tell whether MN is an altitude, a perpendicular bisector, both, or neither.
Example: A balalaika is a stringed musical instrument that has a triangular body. Balalaikas are commonly
played when performing Russian songs and dance music. A three-stringed balalaika is shown. Tell whether
string B is an altitude, a perpendicular bisector, both, or neither.
6.3 – Angle Bisectors of Triangles
Angle Bisector:
Special Segments in Triangles
Segment
Type
Property
Examples:
6.4 – Isosceles Triangle
Isosceles Triangle:
Theorem 6.2:
Theorem 6.3:
Examples: Find the values of the variables
Theorem 6.4:
Example:
Theorem 6.5:
6.5 – Right Triangles
Right Triangle:
Theorem 6.6:
Theorem 6.7:
Theorem 6.8:
Postulate 6.1:
Example: Determine whether each pair of right triangles is congruent by LL, HA, LA, or HL. If not
possible, write not possible.
6.6 – The Pythagorean Theorem
Theorem 6.9 – Pythagorean Theorem:
Example: Find the length of the hypotenuse of the right triangle.
Example: Find the length of one leg of a right triangle if the length of the
hypotenuse is 4 meters and the length of the other leg is 3 meters.
Example: Find the missing measures.
Example: Find the rafter length for a roof that has a 10-foot rise and a 20-foot run.
Theorem 6.10:
Example: The lengths of three sides of a triangle are 4, 5, and 6 meters. Determine whether this triangle is a
right triangle.
Example: The lengths of three sides of a triangle are 10, 24, and 26. Determine whether this triangle is a
right triangle.
6.7 – Distance on the Coordinate Plane
Theorem 6.11:
Example: Use the Distance Formula to find the distance between A(6, 2) and B(4, -4). Round to the nearest
tenth if necessary.
Example: Use the Distance Formula to find the distance between M(0, 3) and N(0, 6). Round to the nearest
tenth, if necessary.
Example: Use the Distance Formula to find the distance between G(-3, 4) and H(5, 1). Round to the nearest
tenth, if necessary.
Example: Determine whether DEF with vertices at D(-2, 2), E(6, 2), and F(2, -2) is isoscleles.
Example: Determine whether ABC with vertices A(-3, 2), B(6, 5), and C(3, -1) is isosceles.
Example: Akio took a ride in a hot-air balloon. The flight began 4 miles north of his house. The balloon
landed 3 miles south and 2 miles east of his house. If the balloon traveled in a straight line between the
starting and ending points of the flight, what was the length of Akio’s balloon ride?