Question

6. There are two parallel lines / and m. Three triangles ABC, PBC and QBC drawn on the same base
BC on line m and the vertices A, P and Q all lie on the line l.
Q
A
P
1
m
C
B
Consider the median and altitude drawn from the vertices A, P and Q for the triangles ABC, PBC and
QBC respectively. Answer the following questions,
(YES/NO)
i) Will all the three altitudes (from vertices A, P and Q) be of same length?
(YES / NO)
ii) Will all the three medians (from vertices A, P and Q) be of same length?​

Answer 1

ANSWER

Let ABC be a triangle AH the height from A to BC and AM the median (M the midpoint between B and C) without loss of generality let's say that H is between B and M so we have ΔBAH and ΔMAH congruent as angles BHA and MHA are equal angle BAH and MAH are equal (per hypothesis of problem) and they share side AH. Therefore BM=HM(orHM=21BM) 

So, Lets' see ΔAHC. AM is bisector therefore we have ratios MCMH=ACAH but MH=21MB=21MC

therefore MCMH=21

So AHC is a right angled triangle at H and AC=2AH. So, sinC=21 and C=300

∠HAC=600 and ∠HAB=∠HAM=21∠HAC=300 So ∠BAC=900

So, ∠BAD=60