Pythagorean Theorem

Summary

This video tutorial covers the Pythagorean theorem, demonstrating its application in solving geometric problems. It explains how to calculate the hypotenuse of a right triangle, find the area of a square given its diagonal, and determine the perimeter of a rhombus and the area of an isosceles trapezoid using the theorem.

Takeaways

• 📐 The Pythagorean theorem is used to solve problems in geometry involving right triangles.
• 📝 The formula for the Pythagorean theorem is \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs.
• 🔍 Example 1: For a right triangle with legs 5 and 12, the hypotenuse \( x \) is calculated as 13.
• 🔢 Example 2: For a right triangle with hypotenuse 10 and one leg 5, the other leg \( y \) is \( 5\sqrt{3} \).
• 🏠 Example 3: To find the area of a square with a diagonal of 12 inches, use the Pythagorean theorem to find the side length and then square it. The area is 72 square inches.
• 🔺 The diagonals of a rhombus bisect each other at right angles.
• 🔄 In a rhombus, all four sides are congruent, and the perimeter can be calculated by determining one side using the Pythagorean theorem and then multiplying by four.
• 🔷 Example 4: For a rhombus with diagonals 14 and 48, each side is 25 units, making the perimeter 100 units.
• 🔻 Example 5: To find the area of an isosceles trapezoid with bases 12 and 20 and legs 5, calculate the height using the Pythagorean theorem and then use the area formula. The area is 48 square units.
• 📏 Important formulas: Area of a square is \( side^2 \), and area of a trapezoid is \( \frac{1}{2}(base1 + base2) \times height \).

Q & A

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It is expressed as \( C^2 = a^2 + b^2 \), where \( C \) is the hypotenuse and \( a \) and \( b \) are the other two sides.

How is the Pythagorean theorem used to find the hypotenuse of a right triangle?

To find the hypotenuse of a right triangle, you can use the formula \( C^2 = a^2 + b^2 \). You substitute the values of the other two sides into the formula, calculate \( C^2 \), and then take the square root of the result to find \( C \).

What is the formula used to calculate the area of a square given its diagonal length?

The formula to calculate the area of a square given its diagonal length \( d \) is \( \text{Area} = \frac{d^2}{2} \). This is derived from the Pythagorean theorem, where the diagonal forms the hypotenuse of two right triangles within the square.

How can you find the perimeter of a rhombus if you know the lengths of its diagonals?

To find the perimeter of a rhombus, you first determine the length of one side using the Pythagorean theorem, since the diagonals bisect each other at right angles. If the diagonals are \( d_1 \) and \( d_2 \), the side length \( s \) is \( \sqrt{\frac{d_1^2 + d_2^2}{2}} \). The perimeter is then \( 4s \).

What is the formula for calculating the area of an isosceles trapezoid?

The formula for calculating the area of an isosceles trapezoid is \( \text{Area} = \frac{1}{2} (b_1 + b_2) \times h \), where \( b_1 \) and \( b_2 \) are the lengths of the two bases and \( h \) is the height of the trapezoid.

How do you calculate the height of an isosceles trapezoid if you know the lengths of the bases and the non-parallel sides?

To calculate the height of an isosceles trapezoid, you can use the Pythagorean theorem on one of the right triangles formed by drawing a height from one base to the other. If the bases are \( b_1 \) and \( b_2 \), and the non-parallel sides are equal, you can set up the equation \( h^2 + \frac{(b_2 - b_1)^2}{4} = \text{side}^2 \) and solve for \( h \).

Why is the Pythagorean theorem useful in solving geometry problems?

The Pythagorean theorem is useful in solving geometry problems because it relates the lengths of the sides of a right triangle, allowing you to find unknown side lengths, calculate areas of shapes like squares and rhombuses, and determine perimeters of shapes like rhombuses.

What is the relationship between the diagonals of a rhombus?

In a rhombus, the diagonals bisect each other at right angles and are perpendicular to each other. This means that each diagonal cuts the other into two equal parts, forming four congruent right triangles within the rhombus.

How can you simplify the square root of 75 when calculating the area of a square with a diagonal of 12 inches?

To simplify the square root of 75, you can factor 75 into \( 25 \times 3 \). Since the square root of 25 is 5, the square root of 75 simplifies to \( 5\sqrt{3} \).

What is the perimeter of a rhombus with diagonals measuring 7 and 24 units?

Using the Pythagorean theorem, the side length \( s \) of the rhombus is \( \sqrt{576 + 49} = \sqrt{625} = 25 \). The perimeter is \( 4 \times 25 = 100 \) units.

Outlines

00:00

📐 Introduction to the Pythagorean Theorem

This paragraph introduces the Pythagorean theorem, a fundamental principle in geometry used to calculate the lengths of sides in right triangles. The formula is presented as \( C^2 = A^2 + B^2 \), where \( C \) is the hypotenuse and \( A \) and \( B \) are the other two sides. The explanation includes a step-by-step process to find the hypotenuse when the lengths of the other two sides are known, using the example of a triangle with sides of 5 and 12. The concept is further applied to solve for an unknown side when the hypotenuse and one other side are given, illustrated with an example where the hypotenuse is 10 and one side is 5. The paragraph also touches on how to simplify square roots and introduces the application of the theorem in word problems, such as calculating the area of a square with a given diagonal.

05:02

📏 Applying Pythagoras in Geometric Shapes

This paragraph delves into applying the Pythagorean theorem to various geometric shapes, starting with a square. The diagonal of the square is used to calculate the side length, which in turn helps determine the area of the square. The explanation demonstrates how to use the theorem to find the side length when the diagonal is known, using the example of a square with a diagonal of 12 inches. The paragraph then moves on to a rhombus, explaining that the diagonals bisect each other at right angles and are used to calculate the side length of the rhombus, which is then used to determine the perimeter. Lastly, the paragraph discusses an isosceles trapezoid, where the theorem is used to find the height, which is necessary for calculating the area. The examples provided include a rhombus with diagonals of 7 and 24 and an isosceles trapezoid with bases of 12 and 20.

10:04

📐 Calculating the Area of an Isosceles Trapezoid

This paragraph focuses on calculating the area of an isosceles trapezoid. It begins by outlining the formula for the area, which is \( \frac{1}{2}(b_1 + b_2) \times h \), where \( b_1 \) and \( b_2 \) are the lengths of the bases and \( h \) is the height. The explanation involves determining the height by creating two right triangles within the trapezoid and applying the Pythagorean theorem. The example provided uses a trapezoid with bases of 12 and 20 and shows how to find the height by setting up an equation based on the lengths of the bases and the hypotenuse. Once the height is determined, the area of the trapezoid is calculated using the previously mentioned formula, resulting in an area of 48 square units.

Keywords

💡Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is central to the video's theme, as it is used to solve various geometric problems involving right triangles. For example, the script uses the theorem to calculate the hypotenuse of a triangle with sides of 12 and 5, demonstrating its practical application.

💡Right triangle

A right triangle is a triangle that has one angle that measures 90 degrees. This type of triangle is crucial in the video because the Pythagorean theorem specifically applies to it. The script discusses how to use the theorem to find the hypotenuse of a right triangle, which is the longest side and forms the right angle. This concept is foundational for solving the problems presented in the video.

💡Hypotenuse

In a right triangle, the hypotenuse is the side opposite the right angle and is the longest side of the triangle. The script emphasizes the importance of the hypotenuse in the context of the Pythagorean theorem, as it is the side that is calculated using the theorem. For instance, the video calculates the hypotenuse of a triangle with sides of 12 and 5, showing how the theorem can be applied to find the length of the hypotenuse.

💡Legs

In a right triangle, the legs are the two sides that form the right angle. The script refers to these sides as 'a' and 'b' when applying the Pythagorean theorem. Understanding the concept of legs is essential for using the theorem correctly, as the lengths of the legs are squared and added together to find the length of the hypotenuse. The video demonstrates this by using a triangle with sides of 12 and 5, where either side can be considered a leg.

💡Square root

The square root of a number is a value that, when multiplied by itself, gives the original number. In the context of the video, the square root is used to solve for the length of the hypotenuse after applying the Pythagorean theorem. The script shows how to take the square root of the sum of the squares of the legs to find the hypotenuse, such as when calculating the hypotenuse of a triangle with sides of 12 and 5.

💡Area

The area of a shape is a measure of the amount of space it occupies. In the video, the concept of area is used to calculate the area of a square and an isosceles trapezoid. The script explains how to use the Pythagorean theorem to find the side length of a square given its diagonal, and then uses this side length to calculate the area. This demonstrates the practical application of geometry in determining the area of shapes.

💡Square

A square is a quadrilateral with all four sides equal in length and all four angles equal to 90 degrees. The video script discusses calculating the area of a square using the Pythagorean theorem, specifically when the diagonal length is known. The script shows how to find the side length of the square and then uses this to calculate the area, illustrating the relationship between the diagonal and the sides of a square.

💡Rhombus

A rhombus is a type of quadrilateral where all four sides have equal length, but the angles are not necessarily 90 degrees. The video script uses the properties of a rhombus, specifically that its diagonals bisect each other at right angles, to calculate the perimeter. The script demonstrates how to use the Pythagorean theorem in the context of a rhombus by finding the length of one side given the lengths of the diagonals.

💡Diagonals

In a square or a rhombus, diagonals are the lines that connect opposite vertices of the shape. The script explains that the diagonals of a rhombus bisect each other at right angles, which is a key property used in the video to calculate the perimeter of the rhombus. The diagonals are also mentioned in the context of a square, where they are used to find the side length using the Pythagorean theorem.

💡Isosceles trapezoid

An isosceles trapezoid is a quadrilateral with a pair of parallel sides (bases) and non-parallel sides (legs) that are equal in length. The video script discusses calculating the area of an isosceles trapezoid using the formula (Area = (1/2) (b1 + b2) * h), where (b1) and (b2) are the lengths of the bases and (h) is the height. The script uses the Pythagorean theorem to find the height of the trapezoid, demonstrating the application of geometry in solving more complex problems.

Highlights

Introduction to the Pythagorean theorem and its application in geometry problems.

Explanation of the formula C² = a² + b², where C is the hypotenuse and a and b are the legs of a right triangle.

Example calculation: Given sides a = 12 and b = 5, finding the hypotenuse C using the Pythagorean theorem.

Calculation of the hypotenuse C when C² = 144 + 25, resulting in C = 13.

Second example: Given hypotenuse C = 10 and side b = 5, finding the other leg a using the theorem.

Calculation of leg a when 10² = a² + 5², resulting in a = √75 simplified to 5√3.

Word problem involving the area of a square with a diagonal of 12 inches.

Use of the theorem to find the side length x of the square, leading to the area calculation.

Calculation of x when 12² = 2x², resulting in x = √72 simplified to 6√2.

Calculation of the area of the square as x² = 72.

Word problem on calculating the perimeter of a rhombus with diagonals of lengths 7 and 24.

Understanding that the diagonals of a rhombus bisect each other at 90 degrees.

Use of the Pythagorean theorem to find the side length s of the rhombus.

Calculation of s when 24² + 7² = s², resulting in s = 25.

Calculation of the rhombus perimeter as 4s = 100 units.

Word problem on finding the area of an isosceles trapezoid with bases of lengths 12 and 20.

Understanding that in an isosceles trapezoid, the non-parallel sides are congruent.

Use of the Pythagorean theorem to find the height H of the trapezoid.

Calculation of H when 5² = 4² + H², resulting in H = 3.

Final calculation of the trapezoid area using the formula (1/2)(b1 + b2)H = 48.

Transcripts

in this video we're gonna go over the

Pythagorean theorem and we're gonna talk

about how to use it to solve problems

associated with geometry so let's go

over the formula first so if we have a

right triangle and this side is called a

B and hypotenuse to C then the formula

is C squared is equal to a squared plus

B squared C is the hypotenuse it's the

longest of the three sides and a and B

are known as the legs of the right

triangle so let's say if we have a

triangle that looks like this and let's

say this side is five and this is 12

calculate the value of x now if you want

to try it feel free to pause the video

so let's use the formula C squared is

equal to a squared plus B squared so a

can be five or twelve it doesn't matter

so let's say if we choose a to be twelve

and B is five the hypotenuse C is across

the box which is X so we have x squared

is equal to twelve squared plus five

squared now twelve times 12 is 144 and

five squared is 25 144 plus 25 is 169

now to calculate the value of x we need

to take the square root of both sides

the square root of 169 is 13 and so

that's how you can calculate the

hypotenuse of a right triangle now let's

work on another example

so let's say the hypotenuse is 10 this

is 5 and our goal is to calculate why go

ahead and do this so let's use the same

formula C squared is equal to a squared

plus B squared so C is the hypotenuse so

in this problem C is 10 a we could say

it's y and then B is 5 so it's gonna be

10 squared is equal to Y squared plus 5

squared 10 times 10 is a hundred and 5

squared is 25 so we need to subtract

both sides by 25 now 100 minus 25 is 75

so now our next step is take the square

root of both sides so Y is equal to the

square root of 75 now how can we

simplify this value to get the right

answer once you get the exact answer and

it's fully simplified for him what

perfect square goes into 75 25 is a

perfect square that goes into it 25

times 3 is 75 and the square root of 25

is 5 so Y is equal to 5 square root of 3

now let's work on some word problems

what is the area of a square with a

diagonal length of 12 inches so first

let's draw a square and so this is just

a rough sketch of a square and this is

the diagonal of the square so that's 12

now let's call this X all 4 sides of a

square are the same so this is X so

notice that we have a right triangle the

area of a square is the left times the

width both the length and the width is

equal to X so the area of a square is x

squared so we can calculate the value of

x we can calculate the area of the

square so let's use the

theorem to calculate X so C squared is

equal to a squared plus B squared C the

hypotenuse is 12 a is equal to X in this

example and B is equal to X so 12

squared is x squared plus x squared 1 x

squared plus 1 x squared is 2x squared +

12 squared is 144 so first we need to

divide both sides by 2 144 divided by 2

is 72 and so that's equal to x squared

now let's take the square root of both

sides so X is equal to the square root

of 72 now a is equal to x squared and x

squared is 72 so a is just 72 so that's

the area of the square by the way if you

want to simplify this radical you can

say that 72 is 36 times 2 and the square

root of 36 is 6 so X is 6 square root 2

but the area is x squared and we can see

that x squared is 72 which means a is 72

in this example number four in rhombus

ABCD b/e is seven and C E is 24

calculate the perimeter of the rhombus

so the first thing needs to know is that

the diagonals of a rhombus bisect each

other and 90 degrees B E is 7 and C E is

24 because of rhombus the diagonals of a

rhombus bisect each other hey E and E C

are congruent be e and E D are congruent

so if B E is 7 e d is 7 and if EC is 24

ie is 24 now all four sides of a rhombus

are congruent

so let's say if we call this s this is s

that's us that's s so the perimeter is 4

s now notice that we have four congruent

right triangles this is 7 this is 24 and

this is s so let's use the Pythagorean

theorem to calculate s C squared is

equal to a squared plus B squared in

this case C is the hypotenuse s a we

could say it's 24 and B is 7 24 times 24

that's 576 and 7 squared is 49

now 576 plus 49 is 625 so now let's take

the square root of both sides the square

root of 625 is 25 so now we can

calculate the perimeter so it's 4 s or 4

times 25 which is a hundred so that's

the perimeter for this particular

rhombus it's a hundred units number five

what is the area of the isosceles

trapezoid shown below

so first we'll need a formula the area

is 1/2 B 1 plus B 2 times H so B 1 is

the first base that's 12 B 2 is the

second base which is 20 and H is the

height of the trapezoid so somehow we

need to calculate H in order to

calculate the area so how can we do so

now the first thing we need to realize

is that for an isosceles trapezoid a B

is congruent to C D so both sides are

equal to 5 next we need to draw two

right triangles now we know that ad is

20 that was given to us in the beginning

now if we add two additional points

let's call this e and f EF is the same

as BC that's 12 now if these two sides

are congruent and this and that has to

be congruent these two a E and F D must

be congruent to each other so if we call

this X and X we could say that X plus 12

plus X is equal to 20 or 2x plus 12 is

equal to 20 so let's subtract both sides

by 12 so 20 minus 12 is 8 and if we

divide by 2 we can see that X is 4

so we could put a four here

so now we can find H the missing side so

now let's focus on this right triangle C

squared is equal to a squared plus B

squared the hypotenuse is 5 a we can say

it's 4 and B is H 5 squared is 25 4

times 4 16 and 25 minus 16 is 9 so H

squared is 9 and if you take the square

root of both sides we can see that H is

equal to 3 so now that we have the value

of H we can calculate the area of the

trapezoid using this formula that is

this formula here so it's going to be

1/2 b1 which is 12 plus B 2 which is 20

times the height of 3 now 12 plus 20 is

32 and half of 32 is 16 16 times 3 is 48

so this right here is the answer that's

the area of the trapezoid