â–³ FGH, J and K are midpoints of FH and HG respectively
1.
Given
2.
FH = FJ+JH and GH = GK+KH
2.
Segment Addition Postulate
3.
FJ = JH and GK = KH
3.
Definition of congruent segments
4.
FH = 2HJ and GH = 2HK
4.
Substitution
5.
FH/HJ = 2 and GH/HK = 2
5.
Division Property of Equality
6.
FH/HJ = GH/HK
6.
Transitive Property of Equality
7.
∠H ≅ ∠H
7.
Reflexive Property of Congruence
8.
â–³ FGH ~ â–³ JKH
8.
SAS Similarity Theorem
9.
∠F ≅ ∠HJK
9.
Definition of similar triangles
10.
JK∥FG
10.
Corresponding Angles Converse
11.
FG/JK = GH/KH
11.
Definition of similar triangles
12.
FG/JK = 2
12.
Substitution
13.
JK=1/2FG
13.
Simplifying
Practice makes perfect
Let's consider â–³ FGH and let J and K be the midpoints of FH and GH, respectively.
Our mission is to prove that JK∥ FG and that JK = 12FG. We will begin to map out these relationships by using the Segment Addition Postulate to write the following pair of equations.
FH = FJ+JH
GH = GK+KH
We can break down this equation further by checking for the relationships expressed on the diagram. By the definition of a midpoint, we know that FJ≅JH and GK≅KH. These relationships imply that FJ=JH and GK=KH, respectively.
FH = 2HJ
GH = 2HK
⇒
FH/HJ = 2 [0.25cm]
GH/HK = 2
Because both proportions equal the same number 2, we can use the Transitive Property of Equality to obtain the following equation.
FH/HJ = GH/HK
Additionally, notice that ∠H is common to both △ FGH and △ JKH. The Reflexive Property of Congruence gives us ∠H ≅ ∠H.
FH/HJ = GH/HK and ∠H ≅ ∠H
In consequence, the Side-Angle-Side (SAS) Similarity Theorem tells us that â–³ FGH ~ â–³ JKH. Therefore, the corresponding angles are congruent.
The Corresponding Angles Converse gives us that JK∥FG. Additionally, the similarity relation gives us the proportions below.
FG/JK = GH/KH = FH/JH
Finally, let's solve the left-hand side equation for JK.
Given: & â–³ FGH, J andK are midpoints of
& FHandHG respectively
Prove: & JK∥FGandJK = 12FG
Let's summarize the proof we did above in the following two-column table.
Statements
Reasons
1.
â–³ FGH, J and K are midpoints of FH and HG respectively
