4. The Triangle Midsegment Theorem
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What characteristics do parallel lines have in a coordinate plane? How can you calculate the length of a segment?
See solution.
To show that DF is parallel to AB, we will show that they have the same slope. Let's begin by determining the endpoints of DF and AB to find their slopes.
| Segment | Points | y_2-y_1/x_2-x_1 | Slope | Simplified Slope |
|---|---|---|---|---|
| DF | D( - 4,-2), F( - 1,- 4) | -4-( -2)/- 1-( - 4) | - 2/3 | -2/3 |
| AB | A( -5, 2), B( 1,- 2) | -2- 2/1-( - 5) | - 4/6 | -2/3 |
From our calculations, we see that DF and AB have the same slope and therefore, they are parallel. Next, we will show that DF= 12AB by calculating the length of DF and AB using the Distance Formula.
| Segment | Points | sqrt((x_2-x_1)^2+(y_2-y_1)^2) | d |
|---|---|---|---|
| DF | D( - 4,-2), F( - 1,- 4) | sqrt(( - 1-( - 4))^2+( -4-( -2))^2) | sqrt(13) |
| AB | A( -5, 2), B( 1,- 2) | sqrt(( 1-( - 5))^2+( -2- 2)^2) | sqrt(52) |
DF= sqrt(13), AB= sqrt(52)
1/b* a = a/b
Rewrite 2 as sqrt(4)
sqrt(a)/sqrt(b)=sqrt(a/b)
Calculate quotient