Can You Solve This Geometry College Entrance Exam From 1904?

Graphic version of the problem as we set it up. Image Credit: IFLScience

The College Entrance Examination Board was founded in 1899 by 12 universities and three high-school preparatory academies. Before it introduced the Standardized Aptitude Test in 1926, the examination looked at students’ knowledge in multiple subjects such as Botany, Chemistry, English, French, German, Greek, History, Latin, Mathematics, Physics, Zoology.

The National Museum of American History has in its collection copies of the College Entrance Exams and we were particularly intrigued by the Mathematics questions from 1904, where 115 years of advances in educations don’t give us too much advantage over the students that sat the actual exam in the past.

Question 3 part b reads: “The sides of a pentagon are respectively 4,5,6,7 and 8 centimeters.  Find the sides of a similar polygon whose area equals four times the area of the given polygon.”

Similar polygons are shapes made of multiple sides in which each corresponding sides are proportional.

Do you think you can solve it? Our solution after this break.


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Our Solution – There are several ways to work out what the sides of the bigger irregular pentagon are. We are looking for the scale factor, the number we need to multiply each side by to get the right bigger pentagon. In this case, a pentagon with an area four times as big.

As we said there are many ways to get to the right solution, but we went for something that was cheap and easy. Mostly we went with what we remembered from geometry in school and it is not too much.

We construct our Pentagon with a right angle between the side of length four and the side of length five. If we connect the two we have a nice easy triangle whose area we can easily calculate (base times height divided by two). So the area of that triangle is 10. In our bigger irregular pentagon, the same area needs to be 4 times bigger. This tells us that each side needs to be twice as long. Our scale factor is 2.

So the sides of the bigger pentagon are respectively 8, 10, 12, 14, and 16 centimeters.

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