How will you draw a square equal in area to a given triangle?

3 Answers
Dec 1, 2017

See below.

Explanation:

Given the triangle ABCABC (light blue) we construct the equivalent area triangle ABC' (pink) by sliding the tip C parallel to the basis AB forming the rectangular triangle ABC'

After that we construct the equivalent area quadrilateral ABDE (blue) with sides bar(AB) and 1/2 bar(AC') and with the same area as triangle ABC

Now we need a square with side L such that L^2= bar(AB) xx 1/2 bar(AC')

and this is done drawing a circumference with origin at 1/2(bar(AB) + 1/2 bar(AC')) and radius r = 1/2(bar(AB) + 1/2 bar(AC')) and drawing a perpendicular line to the diameter issuing from the point (bar(AB),0) and intersecting the circumference at (bar(AB),F). Now the distance from F to the perpendicular foot is L.

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Dec 1, 2017

Find the area of the triangle and then find the square root find the length of the sides of the square.

Explanation:

Assuming that you have all the measurements of the given triangle, you can calculate the area:

A = 1/2bh

If you do not have the measurements, but a scale drawing, you can measure the length of the base and the height.

Once you have the area of the triangle, you also have the area of the square,

To find the length of the side:

s xx s = A

s^2 = A

s = sqrtA

Find the square root of the area to find the length of the sides of the square.

Use this length to draw a square,

This is by calculation, rather than by construction.

Dec 2, 2017

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Steps

  1. The base BC=b of the given DeltaABC is bisected
  2. Perpendicular h from A to BC is drawn.
  3. line segment of length h is cut off from the extended part of BC
  4. Line segment of length b/2+h is bisected and a semicircle is drawn taking b/2+h as diameter
  5. Perpendicular CD on BC is drawn , which intersects semicircle at D. Now the length of CD will be mean proportion of b/2 and h. Hence CD^2=1/2xxbxxh="Area of "Delta ABC
  6. So a square CDEF completed on side CD is the required square.