This High School Geometry Problem has a Simple Solution (it’s just hard to find!)

Let ABCD be a square, and l be a line segment from B to a point on side AD. If A is 5 cm from l and C is 7 cm from l, find the area of ABCD.

All the relevant information is shown in the diagram above. All you need to do is find the area of the square.

Take some time to think about it, before you scroll down to see a hint,or the solution below.

Photo by Daniele Franchi on Unsplash

Hint

Let E be the closest point on line l to A, and point F be the closest point on line l to C. Look at triangles ΔABE and ΔBCF…

Solution

Triangles ΔABE and ΔBCF are congruent (they have the same size and shape). We can realise they are congruent by noticing that they both have:

• a right angle
• a hypotenuse of equal length (because ABCD is a square)
• another common angle
  (because ∠CBF = 90°− ∠ABE
  and using ΔABE gives ∠BAE =…