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# ZingPath: Angles and the Pythagorean Theorem

## Using the Pythagorean Theorem to Solve Problems                Searching for

## Angles and the Pythagorean Theorem

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### Lesson Focus

#### Using the Pythagorean Theorem to Solve Problems

Algebra Foundations

### Learning Made Easy

You will find the shortest path between the top and bottom of a silo using the Pythagorean theorem.

### Now You Know

After completing this tutorial, you will be able to complete the following:

• Use the circumference of the circular base of a cylinder to find the length of its rectangular lateral face.
• Use the Pythagorean theorem to calculate the length of the hypotenuse of a right triangle.

### Everything You'll Have Covered

The Pythagorean theorem: a2 + b2 = c2.

The Pythagorean theorem states:

The sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse. The Pythagorean theorem is used any time we have a right triangle, we know the lengths of two sides, and we want to find the length of the third side.

For example,

Given the right triangle below, if a = 5 units and c = 13 units, what does b equal? Finding the length of the hypotenuse.

The Pythagorean theorem (a� + b� = c�) is used to find the length of the hypotenuse of a right triangle when the lengths of the two legs are known.

For example, Finding the length of the rectangular lateral face of a cylinder.

The circumference of the circular base of a cylinder (2?r) equals the length of its rectangular lateral face. This can be observed in Section 2 of the Activity Object.

For example,

A cylinder has a base with a radius (r) of 10 inches. To find the length (l) of the rectangular lateral face of the cylinder:

The circumference (C) of the circular base = 2?r

r = 10

If we use ? = 3,

C =2 x 3 x 10

= 60 inches

Using the Pythagorean theorem with the net of a cylinder.

In this Activity Object, students will need to use the Pythagorean theorem to find the length of the diagonal of the lateral face of a cylinder, as shown in the diagram below. To find the length of the diagonal, we need to know the height and length of the rectangle. The circumference of the circle (2?r) is equal to the length of the rectangle.

Using the Pythagorean theorem to find the length of the diagonal, we can use the equation, d� = h� + l�.

For example,

For a given cylinder, if h = 16 m and r = 5 m, what is the length of the diagonal?

1.��� First find the length of the rectangle, which is equal to the circumference of the circle (2?r). Let ? = 3.

l = 2 � ? � 5

= 30 m

2.��� Then find the length of the diagonal using the Pythagorean theorem.

d� = (16 m)� + (30 m)�

d� = 256 m� + 900 m�

d� = 1156 m�

d = 34 m

The following key vocabulary terms will be used throughout this Activity Object:

circumference - the complete distance around a circle

diagonal - a line segment connecting two non-adjacent vertices of a polygon

height - the perpendicular distance to the base

hypotenuse - the side of a right triangle opposite the right angle; it is the longest side

length - the distance along a line or shape from one point to another

net - a two-dimensional pattern of a three-dimensional figure that can be folded to form the figure

pi (?) - the ratio of a circle's circumference to its diameter; approximately 3.14

Pythagorean theorem - the sum of the squares of the lengths of the two legs (a and b) of a right triangle is equal to the square of the length of the hypotenuse (c). This can be expressed by: a2 + b2 = c2

radius - the length of a line segment that connects the center of a circle to any point on the circle

right triangle - a triangle in which one angle of the triangle measures exactly 90o

### Tutorial Details

 Approximate Time 20 Minutes Pre-requisite Concepts Learners should be familiar with the circumference of a circle, cylinders, Pythagorean theorem, and square roots. Course Algebra Foundations Type of Tutorial Skills Application Key Vocabulary Pythagorean theorem, Pythagoras,