## Brain-Teaser 222: British triangles

**From The Sunday Times, 25th**** July 1965** [link]

British Triangles, naturally, stand on horizontal bases with their points upwards. All their sides, never more than a sensible 60 inches, and their heights measure an exact number of inches. No B.T. is isosceles or right angled.

You can often put more than one B.T. on a common base. On a base of 28 inches 8 B.T.s are erected.

What are their heights?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser222]

## Jim Randell 9:47 am

on27 November 2022 Permalink |If the 3 sides of a triangle are

a, b, c, then the areaAis given by Heron’s formula:And if the height (from side

a) ish, then we also have:Hence the height

hcan be determined from the sidesa, b, c:This Python program considers possible values for the 2nd and 3rd sides of the triangle, and then looks for values where the height is an integer.

It runs in 53ms. (Internal runtime is 1.8ms).

Run:[ @replit ]Solution:The heights of the triangles are 9″ (2 triangles), 15″ (4 triangles), 24″ (2 triangles).The four triangles found by the program are shown below. And they can be mirrored to produce the remaining four triangles.

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## Hugh Casement 7:41 am

on28 November 2022 Permalink |Alternatively we can think of it as two right-angled triangles joined together.

If their bases are d and e, then d² = b² – h² and e² = c² – h².

d + e, or |d – e| in the case of the obtuse-angled triangles, must equal a = 28.

b must not equal c, for then the overall triangle would be isosceles

(that is the case when h = 48, b = c = 50, and d = e = 14).

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## Hugh Casement 9:09 am

on28 November 2022 Permalink |Joined together along the line of their common height h, of course.

A base a = 21 also allows eight resultant triangles, including reflections.

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