Video Transcript
A square has a diagonal 29. What is its area?
We will begin this question by recalling some key properties of a square. We know that the lengths of each side of a square are equal. If we let the length equal 𝐿 units, the area is equal to 𝐿 squared. If we draw on the diagonal of the square of length 𝑑 units, we create a right-angled triangle. We recall that the Pythagorean theorem states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side, known as the hypotenuse. The diagonal is the hypotenuse. Therefore, 𝐿 squared plus 𝐿 squared is equal to 𝑑 squared. Simplifying the left-hand side gives us two 𝐿 squared. Therefore, two 𝐿 squared is equal to 𝑑 squared.
We can now use these two equations and the fact the diagonal is equal to 29 to calculate the area. As 𝑑 is equal to 29, two 𝐿 squared is equal to 29 squared. 29 squared is equal to 841. Therefore, two 𝐿 squared equals 841. We can divide both sides of this equation by two such that 𝐿 squared is equal to 420.5. Whilst we might be tempted to square root both sides of this equation, we notice that the area is equal to 𝐿 squared. This means that the area is equal to 420.5. Whilst there were no units given in this question, our units for area would be square units.
