The triangles do not have to be assumed to be any type of special triangle.
The question is testing two concepts:
(1) Mid point theorem
(2) Basic Proportionality Theorem
Which are related and similar:
(1st) midpoint theorem states that if we draw a line segment from the midpoint of one side of a triangle across to the midpoint of the opposite side of the triangle ———> then that segment will be the “Mid-segment” of the triangle.
The Mid-Segment will be:
Parallel to the 3rd side
And
Mid segment length = (1/2) * (length of that 3rd side)
This is exactly what is going on with line PQ
PQ is parallel to BC
And
PQ = (1/2) (length of BC) (which is important for the end
The basic proportionality theorem says that the line drawn parallel (line PQ) to one side of a triangle and drawn from mid-point to mid-point will create two similar triangles where:
AC / AP = AB / AQ = 2 / 1
The heights of the similar triangles will be in a 2/1 ratio as well
The same thing is occurring in triangle RPQ with line GH and line PQ
If we use PQ as the base of triangle PQR and BC as the base of triangle ABC we know that the ratio of bases will be:
(Base BC) / (Base PQ) = (2) / (1)
What
chetan2u is doing is drawing a perpendicular height from vertex A to base BC and another perpendicular height from vertex R to base PQ.
Then he proves that these 2 heights are equal - since there exists only one perpendicular distance between 2 parallel lines, the parallel lines of PQ and BC
Since the heights drawn from their respective vertices are equal, the Area of the triangle ABC - to - Area of triangle PQR will be in the same Ratio as ———> (Base BC) / (Base PQ) = (2) / (1)
flipping it around to match the question the answer is
1/2
Note: this is a really hard question. I believe there are 2 official questions that test the Concept of the Midpoint Theorem (one DS and one PS)
I can’t imagine this question would be anything less than an upper 700 type level of question.
CEO2021 wrote:
chetan2u wrote:
Bunuel wrote:
In the figure above P and Q are the mid points of AC and AB. Also, PG = GR and HQ = HR. What is the ratio of the area of triangle PQR to the area of triangle ABC?
A. 1/2
B. 3/5
C. 2/3
D. 4/5
E. 5/6
Take triangle ABC Since Q and P are the midpoints of AC and AB, ABC and AQP are similar triangles with the ratio of sides as 1:2.
Therefore, 2*QP=BC
If the altitude AT=2a, then a is the distance between lines GH and PQ.
Area of triangle ABC = \(A_{ABC}=\frac{1}{2}*BC*AT=\frac{1}{2}*2*QP*2a=2*PQ*a\)
Now, take triangle PQR Since G and H are the midpoints of PR and QR, RGH and RQP are similar triangles with the ratio of sides as 1:2.
Therefore, 2*QP=BC
Since a is the distance between lines GH and PQ, the altitude RS=2*a=2a
Area of triangle PQR = \(\frac{1}{2}*PQ*RS=\frac{1}{2}*
2*QP*2a=\frac{1}{2}*A_{ABC}\)
SO ratio i s1/2
A
chetan2uthx for soln.
above highlighted part should be 1/2*BC
but isn't we are assuming triangle be a special one?
triangle can be as per attached below also...
Pls suggest
thnx
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