midpoint formula
x₁+ x₂/ 2 or y₁+ y₂/ 2
equidistant
if a point is the same distance from the objects
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
Converse of the perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector
Define Concurrent
When three or more lines intersect at one point
Define The point of concurrency
The point at which three lines or more intersect
For any triangle, certain sets of lines are ALWAYS concurrent? True or False
TRUE
Define Concurrency of Perpendicular Bisectors Theorem
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
Define Circumcenter of a triangle
The point of concurrency of the perpendicular bisectors of a triangle
What does it mean to state that a circle is "circumscribed about" the triangle?
A circumcenter is equidistant from the vertices and you can use the circumcenter as the center of the circle that contains each vertex of the triangle
Can a circumcenter be inside, on, or outside a triangle?
yes
Incenter of the triangle
the point of concurrency of the angle bisectors of a triangle
Is the incenter ALWAYS inside a triangle?
yes
Concurrency of Angle Bisectors Theorem
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle
A_________ of a triangle is a segment from a vertex of the triangle to the midpoint of the side opposite the vertex? [centroid, median]
median
The length of the perpendicular segment from a point to a line is the ________? [midsegment, distance from a point to the line]
distance from a point to a line
a MIDSEGMENT OF A TRIANGLE is a segment that connects the midpoint of three sides? [true/false]
false (connects the midpoiont of 2 sides)
Define the MIDSEGMENT THEOREM
DE= 1/2 AB
T is the midpoint of . U is the midpoint of . m∠QUT = 85 and RS = 36.a. Find m∠QSR.b. Find TU.
a. 85; b. 18
DETERMINE THE HEIGHT OF THE BUILDING IN #2
100 FT
FIND THE VALUE OF X IN #3
52
B is the midpoint of AC, D is the midpoint of CE , and BD = 14. Find AE
28
pic#6 Given: CD is the perpendicular bisector of IK . Which statement is true? ∠ICK is a right angle. IJ≅JK D is the midpoint of JK. DK = CK
IJ≅JK
PIC#7Given: DE is the perpendicular bisector of HJ. Which statement is true?
DE ⊥ HJ
PIC#8 Given: CD is the ⊥ bisector of KM . What can you NOT conclude?
CL ≅ LD
PIC#9 Given: JK is the perpendicular bisector of OP . What can you NOT conclude?
M IS THE MIDPOINT OF JK
PIC#10 Find the center of the circle that you can circumscribe about ΔEFG, when E(2, 6), F(2, 2), and G(8, 2).(4, 4) (4, 5) (4, 3) (5, 4)
5,4
PIC#11 Identify the segment interior to the triangle with respect to the whole figure
perpendicular bisector
PIC#12 In ΔACE, G is the centroid. If BE = 12, find BG and GE.
BG = 4, GE = 8
PIC#13 Identify the segment interior to the triangle with respect to the whole figure
angle bisector
pic#14 Find the length of AB given that DB is a median of the triangle and BC is 20
20