PROVE THAT IN A TRIANGLE THE LINE SEGMENT JOINING THE MID POINTS OF ANY 2 SIDES IS PARALLEL TO 3 - Maths - Areas of Parallelograms and Triangles

Question

PROVE THAT IN A TRIANGLE THE LINE SEGMENT JOINING THE MID POINTS OF
ANY 2 SIDES IS PARALLEL TO 3 RD SIDE AND IS HALF OF IT USING THE
ABOVE THEOREM DO THE FOLLOWING P Q R ARE MID POINTS OF SIDES BC AC
AB OF THE TRIANGLE ABC RESPECTIVELY IF PQ=2 5 CM QR=3 CM RP=3 5 CM
FIND THE LENGTH OF SIDES AB BC CA

Akhil Goyal · Accepted Answer

Consider the
diagram below:
<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/ck-files/ck_55177bb659a10%20png">
Consider △s ABC and ARQ
AQ=12AC and AR=12AB⇒AQAC=ARAB=12Also, ∠A is common to both the triangles
 Therefore:△ABC and △ARQ are similar by SAS propertyFrom this we can say that:∠AQR=∠C, and RQBC=12 (since the ratio of the triangle for two similar triangle is applicable on all sides)Since, these are corresponding angles of the pair of sides: RQ and BC, we have: RQ∥BCHence Proved
Now, given: PQ=2
5cmWe see that side opposite to PQ is AB, thus from above theorem:PQAB=12⇒AB=5cmSimilarly, BC=2(RQ)=6cmand, AC=2(PR)=7cm