# Can You Solve This Geometry College Entrance Exam From 1904?

The College Entrance Examination Board was established in 1899, 12 universities and 3 high-school preparation academies.

Before it introduced the Standardized Aptitude Test in 1926, the test looked at students’ knowledge in multiple subjects such as Botany, Chemistry, English, French, German, Greek, History, Latin, Mathematics, Physics, and Zoology. The National Museum of American History has transcripts of college entrance exams, and we have been particularly interested in math questions since 1904, where 115 years of progress in education does not benefit our real students much. Test in the past.

Question 3 reads Part B, “The sides of a pentagon are 4,5,6,7 and 8 centimeters, respectively. Look for the sides of a similar polygon whose area is four times the area of ​​a given polygon.” A similar polygon is a shape formed by multiple sides, which is proportional to each of the related sides.

Do you think you can solve this, our solution after this break? Our solution – there are different ways to implement what are the sides of the big irregular pentagon. We are looking for the scale factor; we have to multiply each side to get the correct large pentagon. In this case, a Pentagon with four times the territory.

As we said, there are many ways to reach the right solution but we went for something cheap and easy. Most of the time we took what we remembered from geometry in school and it is not too much. We build our pentagon with a right angle between four sides of length and five sides of length. If we connect the two, we have a simple triangle whose area we can easily calculate (base time height divided by two).

Therefore, the area of ​​this triangle is 10. In our larger irregular Pentagon, the same area needs to be 4 times larger. This tells us that each side needs to be twice as long. Our scale factor is two. Thus, the sides of the large pentagons are 8, 10, 12, 14 and 16 centimeters, respectively.