Worksheet: Pythagorean Theorem and Distance Formula

Comprehension worksheet generated from the video "Pythagorean Theorem and Distance Formula".

G.RT.1.1Apply the distance formula, the Pythagorean theorem, and the Pythagorean theorem converse (approximate and exact values, including Pythagorean triples) to solve problems, using algebraic and logical reasoning and mathematical models.

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Worksheet: Pythagorean Theorem and Distance Formula

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Vocabulary

Pythagorean Theorem — A theorem stating that in a right triangle, the square of the hypotenuse ( $c^2$ ) is equal to the sum of the squares of the two legs ( $a^2 + b^2$ ).
Right Triangle — A triangle that has one angle measuring 90 degrees.
Hypotenuse — The side opposite the right angle in a right triangle, which is always the longest side.
Legs — The two sides of a right triangle that form the right angle.
Cartesian Plane — A two-dimensional coordinate system used to locate points using ordered pairs $(x, y)$ .

Questions

Watch the video and answer the following questions based on the content presented.

According to the video at 0:28, what is the mathematical relationship between the legs a and b, and the hypotenuse c in a right triangle?

When deriving the distance formula at 1:55, what expressions represent the lengths of the horizontal and vertical legs of the right triangle formed by points

(x_1, y_1)

and

(x_2, y_2)

In the first example (3:24), after calculating the distance between points

(2,1)

and

(-1,3)

, what is the final simplified distance obtained?

In the second example (5:14), which point is identified as the vertex where the right angle is located in the triangle formed by

(2,1)

(4,0)

, and

(5,7)

$(2,1)$
$(4,0)$
$(5,7)$
There is no right angle in this triangle.

Explain in your own words how the Pythagorean Theorem is used to derive the Distance Formula for two points on a Cartesian plane. Describe the role of each component (