Points of Concurrency
1 : √3 : 2
30-60-90 triangle ratio?
1 : 1 : √2
45-45-90 triangle ratio?
always
A triangle (always, sometimes, never) has 3 medians.
obtuse
A triangle has side lengths a, b, and c. The longest side length is c. If a² + b² < c², the the triangle is a(n) _____ triangle.
right
A triangle has side lengths a, b, and c. The longest side length is c. If a² + b² = c², the the triangle is a(n) _____ triangle.
acute
A triangle has side lengths a, b, and c. The longest side length is c. If a² + b² > c², the the triangle is a(n) _____ triangle.
sometimes
An altitude is (always, sometimes, never) a median.
Pythagorean Triple
Any three natural numbers (1, 2, 3, ...) that satisfy the Pythagorean Theorem.
16
BC = ?
2
C is the circumcenter. If CD = 10 and CA = 4x + 2, then x = ?
10
C is the circumcenter. If CD = 10, then CB = ?
13
C is the circumcenter. If DF = 13, then BF = ?
90
C is the circumcenter. The measure of angle CGB is ___°.
20.9
Find JZ.
(4, 3)
Find the circumcenter of a triangle with vertices at (0, 0), (8, 0) and (0, 6).
34°
Find the measure of angle DBC if the measure of angle ABD = 68°.
64°
Find the measure of angle TSU.
2
G is the incenter. If FG = 7 and EG = 3x + 1, then x = ?
7
G is the incenter. If FG = 7, then GD = ?
60
G is the incenter. If the measure of angle GAD = 30°, then the measure of angle DAE = __________°.
30
G is the incenter. If the measure of angle GAD = 30°, then the measure of angle GAE = __________°.
2
Given a 30-60-90 triangle, you can find the length of the hypotenuse by multiplying the short leg by ___.
√3
Given a 30-60-90 triangle, you can find the length of the longer leg by multiplying the short leg by ___.
√2
Given a 45-45-90 triangle, you can find the length of a leg by dividing the hypotenuse by ___.
√2
Given a 45-45-90 triangle, you can find the length of the hypotenuse by multiplying either leg by ___.
6
If AB = 12, then DE = ?.
8
If DE = 4, then AB = ?.
16
If PT = 24, then PV = ?
hinge theorem
If QR > UV, then angle T ___ angle S.
10
If SR = 30, then VS = ?
16
If SV = 8, then VR = ?
9
If UV = 3, then QU = ?.
11
If VP = 22, then VT = ?
27
If VR = 18, then SR = ?
Isosceles
In a(n) ___ triangle, the points of concurrency are collinear and do not coincide. a. scalene b. isosceles c. equilateral d. obtuse
Equilateral
In a(n) ___ triangle, the points of concurrency coincide a. scalene b. isosceles c. equilateral d. obtuse
Altitude
Is a median, altitude, angle bisector, or perpendicular bisector shown?
Angle Bisector
Is a median, altitude, angle bisector, or perpendicular bisector shown?
Median
Is a median, altitude, angle bisector, or perpendicular bisector shown?
Perpendicular Bisector
Is a median, altitude, angle bisector, or perpendicular bisector shown?
Circumcircle
Is the circle an incircle or a circumcircle?
Incircle
Is the circle an incircle or a circumcircle?
Centroid
It divides each median into two sections at a 2:1 ratio. a. incenter b. circumcenter c. centroid d. orthocenter
Incenter
It is equidistant from the three sides of a triangle. a. incenter b. circumcenter c. centroid d. orthocenter
Circumcenter
It is equidistant from the three vertices of a triangle. a. incenter b. circumcenter c. centroid d. orthocenter
9.5
ML = ?
5.5
PR = ?
Midsegment
Segment DE is called a ____.
True
T or F. Segment AB is parallel to segment DE.
Centroid
The ___ is the center of gravity or balancing point of a triangle. a. incenter b. circumcenter c. centroid d. orthocenter
Orthocenter
The ___ of a right triangle is located on a vertex. a. incenter b. circumcenter c. centroid d. orthocenter
Circumenter
The ___ of a right triangle is located on the midpoint of its hypotenuse. a. incenter b. circumcenter c. centroid d. orthocenter
Circumcenter
The building contractor for a city wants to build a public restroom in a park equidistant from the three recreation areas. Where should the restroom be located? a. At the centroid b. At the orthocenter c. At the incenter d. At the circumcenter
Euler
The centroid, circumcenter, and orthocenter are collinear. They all lie on a straight line called the ___ line.
Circumcenter
The distance from the centroid to the orthocenter is twice the distance from the centroid to the ___. a. incenter b. circumcenter c. centroid d. orthocenter
Median
The line segment joining a vertex to the midpoint of the opposing side is the ___. a. median b. altitude c. angle bisector d. perpendicular bisector
...
The line segment that is both perpendicular to a side of a triangle and passes through its midpoint is the ___. a. median b. altitude c. angle bisector d. perpendicular bisector
Altitude
The perpendicular segment from a vertex to its opposite side is the ___. a. median b. altitude c. angle bisector d. perpendicular bisector
Angle Bisector
The segment drawn from a vertex that bisects that vertex angle is the ___. a. median b. altitude c. angle bisector d. perpendicular bisector
Orthocenter
The three altitudes of a triangle intersect at the a. incenter b. circumcenter c. centroid d. orthocenter
Incenter
The three angle bisectors of a triangle intersect at the a. incenter b. circumcenter c. centroid d. orthocenter
Centroid
The three medians of a triangle intersect at the a. incenter b. circumcenter c. centroid d. orthocenter
Circumcenter
The three perpendicular bisectors of a triangle intersect at the a. incenter b. circumcenter c. centroid d. orthocenter
Centroid
The triangle center shown is the ___. a. incenter b. circumcenter c. centroid d. orthocenter
Circumcenter
The triangle center shown is the ___. a. incenter b. circumcenter c. centroid d. orthocenter
Incenter
The triangle center shown is the ___. a. incenter b. circumcenter c. centroid d. orthocenter
Orthocenter
The triangle center shown is the ___. a. incenter b. circumcenter c. centroid d. orthocenter
True
True of False. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
True
True or False. Each leg of a right triangle ABC is an altitude of ABC.
False (must be right triangle)
True or False. If a triangle has side lengths that measure 5, 5, and 5 then the side lengths form a Pythagorean Triple.
False (5 + 7 is not > 12)
True or False. The measures 5, 7, and 12 can be the side lengths of a triangle.
False (5 + 7 is not > 14)
True or False. The measures 5, 7, and 14 can be the side lengths of a triangle.
True (6 + 7 > 12)
True or False. The measures 6, 7, and 12 can be the side lengths of a triangle.
False
True or False. The three angle bisectors of a triangle intersect at a point outside the triangle.
True
True or False. The triangle centers are located inside an acute triangle.
17°
V is the incenter. Find the measure of angle VKL.
47°
What is the measure of angle SVU?
Orthocenter and Circumcenter
Which two points of concurrency are always located outside of an obtuse triangle? a. incenter b. circumcenter c. centroid d. orthocenter
Possible Answer: y - 5 = -2(x - 1)
Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints S(-1, 4) and B(3, 6).
