# Lesson 2

Half a Square

- Let’s investigate the properties of diagonals of squares.

### 2.1: Diagonals of Rectangles

Calculate the values of \(x\) and \(y\).

### 2.2: Decomposing Squares

- Draw a square with side lengths of 1 cm. Estimate the length of the diagonal. Then calculate the length of the diagonal.
- Measure the side length and diagonal length of several squares, in centimeters. Compute the ratio of side to diagonal length for each.
- Make a conjecture.

### 2.3: Generalize Half Squares

Calculate the lengths of the 5 unlabeled sides.

Square \(ABCD\) has a diagonal length of \(x\) and side length of \(s\). Rhombus \(EFGH\) has side length \(s\).

- How do the diagonals of \(EFGH\) compare to the diagonals of \(ABCD\)?
- What is the maximum possible length of a diagonal of a rhombus of side length \(s\)?

### Summary

Drawing the diagonal of a square decomposes the square into 2 congruent triangles. They are right isosceles triangles with acute angles of 45 degrees. These congruent angles make all right isosceles triangles similar by the Angle-Angle Triangle Similarity Theorem.

Consider an isosceles right triangle with legs 1 unit long where \(c\) is the length of the hypotenuse. By the Pythagorean Theorem, we can say \(1^2+1^2=c^2\) so \(c=\sqrt2\). The hypotenuse of an isosceles right triangle with legs 1 unit long is \(\sqrt2\) units long.

Now, consider an isosceles right triangle with legs \(x\) units long. By the Angle-Angle Triangle Similarity Theorem, the triangle is similar to the isosceles right triangle with side lengths of 1, 1, and \(\sqrt2\) units. A scale factor of \(x\) takes the triangle with leg length of 1 to the triangle with leg length of \(x\). Therefore, the hypotenuse of the isosceles right triangle with legs \(x\) units long is \(x\sqrt2\) units long.

In triangle \(ABC, x=6\) so \(AC\) is 6 units long and \(BC\) is \(6\sqrt2\) units long.

In triangle \(DEF, x\sqrt2=12\) so \(x=\frac{12}{\sqrt2}\), which means both \(EF\) and \(DF\) are \(\frac{12}{\sqrt2}\) units long.