Unit 5: Triangle Properties
How To Find the Median in any Triangle
Medians are segments that connect a vertex to the midpoint of the opposite side.
Look at what the question is asking you and find those two points in the triangle.
Find the vertex and draw a line to the opposite side midpoint.
Determine the value of x and AB if BQ is a median of triangle ABC, AQ = 5x - 4, QC = 3x + 12, and AB = 4x - 11.
Example of Finding Median
AQ = QC
5x-4 = 3x+12
5x = 3x+18
2x = 18
x = 9
AB = 4(9) - 11
AB = 36 - 11
AB = 25
How To Find the Altitude in any Triangle
Altitudes are segments that are formed by drawing a segment from one side of the triangle to the opposite vertex. It is the shortest distance from to the vertex to the opposite side.
Find the vertex that was given to you.
Draw a line all the wat across to the opposite side to make a 90 degree angle.
Example of Finding Alttitude
SR is an altitude of triangle RST. Solve for x if m<SQR = 5x - 3 and m<TRQ = 4x + 3
m<SRQ + m<TRQ = 90
5x-3 + 4x+3 = 90
5x+4x-3+3 = 90
9x+0 = 90
9x + 90
x = 10
How To Find an Angle Bisector in any Triangle
Angle Bisectors are segments/rays/lines that bisects and angle of the triangle.
Find the angle that is given.
Draw a line that starts at the angle and goes through the opposite side.
m<KJL = m<IJL
x+66 = 9x+2
66 = 8x +2
64 = 8x
8 = x
Example of Finding Angle Bisector
JL is an angle bisector of triangle IJK. m<KJL = x+66, m<IJL = 9x+2, and KJ = 2x-5. Determine x and KJ.
KJ = 2(8)-5
KJ = 16-5
KJ = 11
Perpendicular Bisector Property
How to Find a Perpendicular Bisector in any Triangle
Perpendicular bisectors are segments/rays/lines that pass through the triangle and are perpendicular to one side of the triangle.
Find the midpoint from the segment that you were given.
Draw a line from that midpoint straight up until you pass another side.
Draw a square at the bottom corner because you just made a 90 degree angle.
XY is a perpendicular bisector of PR. Determine PR of m<PXY = 8x+2, and PX = 3x-7.
Example of Finding Perpendicular Bisector
Triangle Inequality Theorem
Triangle Inequality Theorem: the sum of any two sides of a triangle must be greater than the third.
32 + 51 > 20 *
32 + 20 > 51 *
20 + 51 > 32 *
AB + BC > AC
AB + AC > BC
AC + BC > AB
1 + 2 > 3
1 + 3 > 2
2 + 3 > 1
9 + 4 > 3 *
9 + 3 > 4 *
3 + 4 > 9 x
Comparing Angles Using Side Measures and Vice Versa
The mid-size side is opposite the middle angles and vice versa.
The longest side is opposite the biggest angle and vice versa.
The shortest side is opposite the smallest angles and vice versa.
Order the sides from shortest to longest.
SSS and SAS Inequality Theorems
Converse of Hinge Theorem (SSS Inequality):
Suppose you have two pairs of congruents sides in two different triangles.
Hinge Theorem (SAS Inequality):
Start with two pairs of congruent sides in two different triangles.
Included angles are different measures.
Triangle ABC is congruent to Triangle DEF.
Included angles are congruent.
Inequalities Between 2 Trinangles
Compare the angles to the non-congruent sides.