Given the isosceles triangle $ABC$ with $| AB | = | AC | = 10$ and $| BC | = 15$. Let points $P$ in $BC$ and $Q$ in $AC$ chosen such that $| AQ | = | QP | = | P C |$. Calculate the ratio of areas of the triangles $(PQA): (ABC)$.
Problem
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Tags: ratio, geometry, equal segments, isosceles
samrocksnature
17.02.2021 08:59
What does the absolute value sign mean? Are we dealing with quantum theory here?
natmath
17.02.2021 09:08
Scale the triangle by $\frac{2}{5}$ so that $AB=4$ and $BC=6$.
Clearly $\Delta PQC\sim \Delta ABC$
If $PQ=x$, then we have
$$x+\frac{3}{2}x=4$$$$x=\frac{8}{5}$$If $\theta=\angle ACB$, we have that
$$[PQA]=\frac{1}{2}x^2\sin\theta$$$$[ABC]=12\sin\theta$$So the ratio is $\frac{x^2}{24}=\boxed{\frac{8}{75}}$
JustinLee2017
17.02.2021 09:13
samrocksnature wrote: What does the absolute value sign mean? Are we dealing with quantum theory here? Lengths.
a.shyam.25
17.02.2021 19:04
Is it necessary though? When has length been negative?
samrocksnature
17.02.2021 19:08
a.shyam.25 wrote: Is it necessary though? When has length been negative? Quantum go brrrr
natmath
17.02.2021 19:09
@above it's simply notation for the length of a segment. It doesn't actually mean absolute value