provide an example where the concept of similar triangles helped solve a problem

Answer 1

In 1943 a squadron of Lancaster bombers staged one of the most audacious raids in history - destroying two gigantic dams in Germany's industrial heartland and cutting the water supply to arms factories - with a revolutionary bouncing bomb invented by British engineer Barnes Wallis. Wallis and the pilots of 617 Squadron - a lively mix of Britons, Australians, Americans, and Canadians - were hailed as heroes and dealt a mighty blow to the German war machine.

The bombs had to be dropped at very low altitude so they would "skip" across the water over the nets that protected the dam, rotating backwards so that they would "hug" the dam as they sank before they exploded -- AND THEY MUST BE RELEASED AT THE BEST DISTANCE !!

The bombadier used a "sling shot" - like stick for which they knew the distance between the two posts. Holding the sling shot at arm's length (known), when the two two towers of the Dam (distance known from aerial photographs), the bomb should be released. [note: in darkness and while being shot at]..

The triangle formed by the eye and the two posts on the slingshot is similar to the triangle formed by the eye and the two towers on the Dam.

See the PBS NOVA video "Bombing Hitler's Dams."

Answer 2

The usual example is measuring the height of a tree (building, column, flagpole, etc) by measuring the shadow cast by an object of known height and the shadow of the object to be measured along with the distance of the tip of the shadow to the base of the object.

The use of similar triangles for this kind of purpose is an example of the use of proportion.