Conjectures

By the Triangle Midsegment Theorem, we know that the midsegment between two sides of a triangle has the following properties.

The midsegment is parallel to the third side.
The midsegment is $\frac{1}{2}$ of the third side.

Now we can make conjectures about a quarter segment and eight segment.

Segment	Conjecture
Quarter	Parallel to the third side and $\frac{3}{4}$ of the third side's length
Eight	Parallel to the third side and $\frac{7}{8}$ of the third side's length

Testing our conjectures

To investigate if our conjectures are true, we need to know the coordinate's of these segments. We know that $L$ is at the origin. Let's call the coordinates of the two remaining vertices $M (8 q, 8 r),$ and $N (8 p, 0) .$

The midsegment $\overline{X Y},$ will divide the triangle's sides in two equal halves. Since we now have the coordinates of $L, M,$ and $N,$ we can determine $X$ and $Y$ using the Midpoint Formula.

Point	Side	Points	$M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	Midpoint
$X$	$\overline{M L}$	$L (0, 0),$ $M (8 q, 8 r)$	$M (\frac{0 + 8 q}{2}, \frac{0 + 8 r}{2})$	$M (4 q, 4 r)$
$Y$	$\overline{M N}$	$M (8 q, 8 r),$ $N (8 p, 0)$	$M (\frac{8 q + 8 p}{2}, \frac{8 r + 0}{2})$	$M (4 q + 4 p, 4 r)$

We will add the coordinates of $X$ and $Y$ to the diagram.

Note that $D$ and $E$ are the midpoints of $\overline{L X}$ and $\overline{N Y},$ respectively. Therefore, we can find the coordinates of $D$ and $E$ by using the Midpoint Formula.

Point	Side	Points	$M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	Midpoint
$D$	$\overline{M L}$	$L (0, 0),$ $M (4 q, 4 r)$	$M (\frac{0 + 4 q}{2}, \frac{0 + 4 r}{2})$	$M (2 q, 2 r)$
$E$	$\overline{M N}$	$M (4 q + 4 p, 4 r),$ $N (8 p, 0)$	$M (\frac{( 4 q + 4 p ) + 8 p}{2}, \frac{4 r + 0}{2})$	$M (2 q + 6 p, 2 r)$

We will add these coordinates to the diagram.

Finally, we have to find the coordinates of the eighth segment. Just like $D$ and $E$ were the midpoints of $\overline{L X}$ and $\overline{N Y},$ the endpoints of the eight segment will be the midpoints of $\overline{L D}$ and $\overline{N E} .$ Since we have the coordinates of $D$ and $E$ , we can calculate these points which we will call $F$ and $G .$

Side	Points	$M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	Midpoint
$\overline{L D}$	$L (0, 0),$ $M (2 q, 2 r)$	$M (\frac{0 + 2 q}{2}, \frac{0 + 2 r}{2})$	$M (q, r)$
$\overline{N E}$	$M (2 q + 6 p, 2 r),$ $N (8 p, 0)$	$M (\frac{( 2 q + 6 p ) + 8 p}{2}, \frac{2 r + 0}{2})$	$M (q + 7 p, r)$

We will add these coordinates to the diagram.

When we know the coordinates of all points, we can immediately see that the quarter segment and eight segment are both parallel to the third side. We can say this because they are all drawn between endpoints with the same $y -$ coordinates which means they are all horizontal.

Finally, we also have to determine the length of our segments. Since the segments are horizontal, we can by the Ruler Postulate determine the distance of these segments by calculating the absolute value of the difference of the endpoints

x -

coordinates.

LN = ∣ 8 p - 0 ∣ = 8 p FG = ∣ (q + 7 p) - r ∣ = 7 p DE = ∣ (2 q + 6 p) - r ∣ = 6 p

Now we can determine the ratio of the quarter and eight segment to the third side by dividing their lengths with

\overline{L N} .

\frac{D E}{L N} = \frac{6 p}{8 p} = \frac{3}{4} \frac{F G}{L N} = \frac{7 p}{8 p} = \frac{7}{8}

As we can see, the second part of our conjecture is also true.

Exercise

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