Question #ffc84

2 Answers
Apr 22, 2016

Answer:

At some point measurement has to be made. The calculations below
Gives the # "base "->z_t=(2xx11)/(3+sqrt(3)) ~~4.649# to 3 decimal places.

Explanation:

The sum of the internal angles on a triangle is #180^o#. As the two given angles sum to #90^o# the remaining angle is also #90^o#. So we have a right triangle. Not only that, the angle given match those found in 1/2 of an equilateral triangle. So we have:

TonyB

By the properties of similar triangles and ratio of proportionality we can now determine the length of the sides

By ratio

#(z_t+o_t+z_t)/(a+b+c) = z_t/c =y_t/a=o_t/b#

#=>11/(2+1+sqrt(3))=z_t/2 = y_t/sqrt(3)=o_t/1#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#=> "base "->z_t=(2xx11)/(3+sqrt(3)) ~~4.649# to 3 decimal places

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#=> o_t= 11/(3+sqrt(3))~~2.325 # to 3 decimal places

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#=>y_t=(11xxsqrt(3))/(3+sqrt(3)) ~~4.026 # to 3 decimal places

May 6, 2016

Answer:

I discuss here a common method
It is not restricted to right angled triangle only.It is applicable when the given angles can be drawn with the help of compass and ruler.

Explanation:

selfsrawn

Construction

  • A straight line #PQ=11 cm# is first drawn.
  • With the help of a pencil compass and a ruler then we draw #/_YPQ=15^@ "and"/_YQP = 30^@#.As a result PY and QY intersect at Y
  • Again with the help of pencil compass and a ruler we draw #/_PYO=15^@ "and"/_QYZ = 30^@#. As a result PO intersects PQ at O and YZ intersects PQ atZ
  • #DeltaYOZ# is the required triangle drawn.

Proof

  • In #DeltaYPO,/_OYP=15^@=/_OPY# by construction, So #DeltaYPO# is an isosceles triangle whose #YO =PO#
  • In #DeltaZYQ,/_ZYQ=30^@=/_ZQY# by construction, So #DeltaYPO# is an isosceles triangle whose #YZ =ZQ#
  • Now in #DeltaYOZ,/_YOZ=/_OYP+/_OPY=30^@,"since"/_YOZ" is the exterior angle of" Delta PAB #,
  • Also in #DeltaYOZ,/_YZO=/_ZYQ+/_YQZ=60^@,"since"/_YZO" is the exterior angle of" Delta QYZ #,
  • Hence in #DeltaYOZ,/_YOZ=30^@,/_YZO=60^@ "and Perimeter=" YO+OZ+ZY = PO+OZ+ZQ=PQ =11 cm#