Pythagorean Theorem
Summary
TLDRThis 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
📐 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.
📏 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.
📐 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.
Mindmap
Keywords
💡Pythagorean theorem
💡Right triangle
💡Hypotenuse
💡Legs
💡Square root
💡Area
💡Square
💡Rhombus
💡Diagonals
💡Isosceles trapezoid
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
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