Question

triangle midsegment theorem:

• triangle midsegment theorem: if a segment joins the ______ of two sides of a triangle, then the segment is ____ to the third side and is ______.

if... ( dg cong ge ) and ( fh cong he ) then... ________
and ________

1. find ( ca ).

1. find ( qp ).

1. if ( u, v, ) and ( w ) are the midpoints of the sides of ( \triangle xyz ) and ( xy = 42, uw = 33, xz = 50, ) then find the indicated measures.

( yz = ) ________
( uv = ) ________
( wv = ) ________
( vz = ) ________

solve for ( x ).
( x = ) ________

triangle angle bisector theorem:

• triangle - angle - bisector theorem - if a ray ______ an angle of a triangle, then it divides the ____ into two segments such that the ratio ____ is the ____ as the ratio ______ adjacent to each segment.

if... ( angle uvx cong angle wvx ) then... ________

find the length of each segment.
( overline{ef} ) and ( overline{fg} )
( overline{rv} ) and ( overline{tv} )

Explanation:

Step1: State Midsegment Theorem

If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half the length of the third side.
For $\triangle DEF$:
$GH \parallel DF$
$GH = \frac{1}{2}DF$

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Step2: Solve for $CA$

$FG$ is midsegment, so $CA = 2 \times FG$
$CA = 2 \times 7 = 14$

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Step3: Solve for $QP$

$QP$ is midsegment, so $QP = \frac{1}{2}VX$
$QP = \frac{1}{2} \times 8 = 4$

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Step4: Solve $\triangle XYZ$ midsegments

1. $UW$ is midsegment for $YZ$: $YZ = 2 \times UW = 2 \times 33 = 66$
2. $UV$ is midsegment for $XZ$: $UV = \frac{1}{2}XZ = \frac{1}{2} \times 50 = 25$
3. $WV$ is midsegment for $XY$: $WV = \frac{1}{2}XY = \frac{1}{2} \times 42 = 21$
4. $V$ is midpoint of $YZ$: $VZ = \frac{1}{2}YZ = \frac{1}{2} \times 66 = 33$

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Step5: Solve for $x$ (midsegment)

$PQ$ is midsegment, so $4x-34 = 2(x-1)$
Expand: $4x - 34 = 2x - 2$
Rearrange: $4x-2x = 34-2$
Calculate: $2x=32 \implies x=16$

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Step6: State Angle Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments such that the ratio of the two segments is the same as the ratio of the other two sides adjacent to each segment.
For $\triangle UVW$: $\frac{UX}{XW} = \frac{VU}{VW}$

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Step7: Solve $EF$ and $FG$

By Angle Bisector Theorem: $\frac{EF}{FG} = \frac{ED}{DG}$
$\frac{x}{x+2} = \frac{8}{12}$
Simplify $\frac{8}{12}=\frac{2}{3}$, cross multiply: $3x=2(x+2)$
Expand: $3x=2x+4 \implies x=4$
$EF=4$, $FG=4+2=6$

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Step8: Solve $RV$ and $TV$

By Angle Bisector Theorem: $\frac{RS}{SV} = \frac{RV}{TV}$
$\frac{40}{16} = \frac{3y}{y+3}$
Simplify $\frac{40}{16}=\frac{5}{2}$, cross multiply: $5(y+3)=2(3y)$
Expand: $5y+15=6y \implies y=15$
$RV=3 \times 15=45$, $TV=15+3=18$

Answer:

Triangle Midsegment Theorem:

Midpoints; parallel; half the length
$GH \parallel DF$; $GH = \frac{1}{2}DF$

1. $CA = 14$
2. $QP = 4$
3. $YZ=66$, $UV=25$, $WV=21$, $VZ=33$

Solve for $x$: $x=16$

Triangle Angle Bisector Theorem:

bisects; opposite side; of the two segments; same; of the other two sides
$\frac{UX}{XW} = \frac{VU}{VW}$
Find segment lengths:
$EF=4$, $FG=6$
$RV=45$, $TV=18$