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The problem is as follows:

Sketch of the problem

The alternatives in my book are as follows:

$\begin{array}{ll} 1.&\textrm{1 cm}\\ 2.&\textrm{3 cm}\\ 3.&\textrm{5 cm}\\ 4.&\textrm{6 cm}\\ \end{array}$

I was only able to spot on:

$\triangle APT$ then $AT= 5$

Then:

$CT=4\cdot5=20$

But that's it. What else to do from here?.

What I attempted to find is some way to get:

$\frac{TO}{TC}=\frac{OR}{QC}$

However none of those segments seem to be known. How can this be solved relying only in euclidean geometry postulates?.

If possible please include a drawing in the answer so I can spot if a construction is needed.

Chris Steinbeck Bell
  • 4,175

2 Answers2

1

Hints:

  • $\triangle APT$ is similar to both $\triangle CQT$ and $\triangle ORT$
  • $O$ is the midpoint of $AC$
  • Try to find the lengths of $AC$ and thus $CO$ and $OT$
  • Then find the lengths of $CQ$ and thus $OR$
Henry
  • 157,058
0

$APT$ right triangle

$AP=4; PT=3$

$AT = 5$ Pythagoras' theorem

$CT=4AT=20\to AC=25$

$OA=25/2=12.5$

$OT=OA-AT=12.5-5=7.5$

Triangles $ATP$ and $OTR$ are similar

$OR/OT=AP/AT$

$OR=AP/AT OT=4/5 7.5$

$OR=6$

Raffaele
  • 26,371