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8/18/2019 JMEV ringbind

1/18

Blue’s Treasure Hunt

on Midline TheoremSTATEGIC INTERVENTION MATERIALMATHEMATICS 9

Jericka Mae E. Valdrez


2/18

Steve gave me this treasure

map and he let me fnd thetreasure. He instructed me thatin every place, there will be afgure wherein I need to fnd themeasurement o either thebase or the midline or me to

move to the next destination.Please help me as we getthrough.


3/18

ARALLEL From ree!" para allelois #beside one another$ Lines are parallel if they lie in the same plane, and are

the same distance apart over their entire length

The Arro!s %o show that lines are parallel, we draw small arrow mar!s

on them. In the fgure below, note the arrows on the linesP& and 'S. %his shows that these lines are parallel. I thediagram has another set o parallel lines they would havetwo arrows each, and so on.

Shorthand Notation(hen we write about parallel lines there is a shorthand

we can use.(e can writewhich is read as #the line segment P& is parallel to thesegment 'S#.

'ecall that the hori*ontal bar over the letters indicates itis a line segment.

P

S'

&

+eore we learn todetermine themeasures o themidline o the

obects by applyingthe midlinetheorem, we mustfrst !now thedefnition o

parallel lines,midpoints andmidline or the mid-segment


4/18

%he mid"se#ment o a polygon also

called a midline/ is a segment oiningthe midpoints o two sides o apolygon.

+eore we learn to

determine themeasures o the

midline o theobects byapplying themidline theorem,we must frst

!now thedefnition oparallel lines,midpoints andmidline or themid-segment

In geometry, the mid$oint is themiddle point o a line segment. Itis e0uidistant rom both endpoints, and it isthe centroid both o the segment and o the

endpoints. It bisects the segment.

midpoint


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%e&nition'

%he mid"se#ment o a triangle also called amidline/ is a segment oining the midpoints otwo sides o a triangle.

ro$erties'

1. %he mid"se#ment o a triangle oins the midpoints o two sides o atriangle such that it is $arallel to the third side o the triangle.

2. %he mid"se#ment o a triangle oins the midpoints o two sides o atriangle such that its length is hal( the len#th o( the third side othe triangle.

3I45I67 %H78'73 86 %'I9657S


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3I45I67 %H78'73 86 %'I9657SE)am$l

es'1. iven DE is the length o the mid-segment. Find AB. Solution'

%he mid-segment is hal o the thirdside.: is hal o 1;.

AB < 1;.

2. iven DE, DF, and FE are the lengths o mid-segments. Find theperimeter o triangle ABC.

Solution' %he mid-segment is hal o

the third side.= is hal o 12 so AC < 12: is hal o 1; so CB < 1;> is hal o 1= so AB < 1=

%he perimeter o the largetriangle ABC is"

12 ? 1; ? 1= < *+,

Here are someexamples o

problems applyingthe midlinetheorem on

t

riangles


7/18

5et us fnd nowthe treasure


8/18

(e must go frst atthe playground.

Particularly to theswing area andsand box area.


9/18


10/18

3I45I67 %H78'73 86 %'9P7@8I4S %he mid-line o a trape*oid is parallel to its bases.

%he length o the mid-line o a trape*oid is hal o the sum o thelengths o its bases.roo( '5et ABC% be a trape*oid with the bases AB and %C and the mid-lineE-

-i#ure ./. 5et us draw the straight line %- through the points %

and - till the intersection with the extension o the straight line AB at thepointG -i#ure +/. Aompare the triangles %-C and -BG.

-i#ure ., %rape*oid and its mid-line-i#ure +, %o the proo o the Theorem


11/18

3I45I67 %H78'73 86 %'9P7@8I4S

-i#ure +. %o the proo o the Theorem

%he segments FA and +F are congruent since the point F is themidpointo the side +A. %he angles 4FA and +F are congruent as thevertical angles.

%he angles 4AF and F+ are congruent as the alternate exteriorangles

at the parallel lines 9+ and 4A and the transverse +A .Hence, the triangles 4FA and F+ are congruent in accordancewith the 9S9-test o congruency o triangles.

It implies that the segments %- and G- are congruent as thecorresponding sides o the congruent triangles %-C and -BG.

-


12/18

3I45I67 %H78'73 86 %'9P7@8I4S

-i#ure +. %o the proo o the Theorem

It is well !nown act that the straight line segment connecting themidpoints o the triangle AG% is parallel to the triangle base AG and its length is hal o the length o the triangle base.

In our case, the length o the segment E- is hal o the lengthAG "

BE-B < BAGB < BABB ? BBGB/.

Since BBGB < B%CB rom the triangles congruency, we haveE-B < BABB ? B%CB/, or BE-B < a ? d/, where a and d are thelengths o the trape*oid bases.

%hus the proo o the Theorem . is ully completed.


13/18

3I45I67 %H78'73 86 %'9P7@8I4SE)am$le

'Find AB.

Solution'

Here is an exampleo problem

applying themidline theorem on

trape*oids


14/18

(e are going nowto the Patio area,to the tree and

lastly to the bushor us to go where

the treasure is.


15/18

EXERCISE 1.2

Find the missing length indicated.

1/ FindWV

2/ FindDG

C/ Find V


16/18

(e ound the treasureD %han! you or helping

me.


17/18

EXERCISE 1.1

1. Find CDSolution'

%he mid-segment is hal o the third side.> is hal o 1=CD < >

2. Find ACSolution'

%he mid-segment is hal o the third side.; is hal o >.

AC < >

!!

ANS/ER 0E1


18/18

ANS/ER 0E1

EXERCISE 1.2

1. Find WV

Solution' %he median is hal the sum o the bases.

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