# Lengths of Squares and Right Triangles

## Learning Goal

Students will explore relationships between areas and side lengths of squares, right triangles, and composite shapes, and learn how to find the length of the hypotenuse and the distance between two points.

## Visual 1: Click and drag the points to explore areas and side lengths of rectangles, squares, and right triangles.

## Question 1: Use the visual to...

- For each square, find the side lengths.
- For each right triangle, find the
**leg**lengths and estimate the**hypotenuse**length.

**Hint:**The area of the movable square is known, and for a square , so .

## Question 1: Check your answer.

## Visual 2: Click and drag the points to explore areas and side lengths of rectangles, squares, and right triangles.

## Question 2: Use the Visual to...

- Find the area of the fixed square.
- Find the side lengths of the fixed square.

**Hint:**Use squares and/or right triangles to compose the fixed square.

## Question 2: Check your answer.

## Visual 3: Click and drag the points to explore areas and side lengths of rectangles, squares, and right triangles.

## Question 3: Use the Visual to...

Find the **hypotenuse** length of Triangle 1.
**Hint:** You will do something similar to what you did in Question 2. Notice that the **hypotenuse** is the side of the inner square.

## Question 3: Check your answer.

## Visual 4: Click and drag the points to explore the distance between them.

## Question 4: Use the Visual to...

Find the distance between each set of two points.

- (2, 2) and (4, 4)
- (-4, 6) and (1, -4)

**Hint 1:**The

**hypotenuse**is the side of the square and the distance.

**Hint 2:**Use a composition of shapes to find the area of the square, so you can find the side length.