### Video Transcript

In this video, we will learn how to
apply the Pythagorean theorem in geometric questions and real-life situations. We will begin by recalling what the
Pythagorean theorem states.

The Pythagorean theorem states that
in any right triangle, the square of the hypotenuse is equal to the sum of the
squares of the two shorter sides. If the hypotenuse is labeled 𝑐 and
the two shorter sides are labeled 𝑎 and 𝑏, then the Pythagorean theorem states
that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. We will now use this theorem to
solve some real-life problems in context.

A man on the top of a building
wants to have a guy wire extended to a point on the ground 20 foot from the base of
the building. To the nearest foot, how long does
the wire have to be if the building is 50 foot tall?

Let’s begin by drawing a
diagram. We are told that the building is 50
foot tall. The wire is extended to a point on
the ground 20 foot from the base of the building. We need to calculate the length of
this wire, which we will call 𝑥. We can see from the diagram that
these values create a right triangle. To calculate the missing length in
any right triangle, we can use the Pythagorean theorem. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or
hypotenuse. The longest side is always opposite
the right angle.

Substituting in our values gives us
20 squared plus 50 squared is equal to 𝑥 squared. The values 20 and 50 could be
written in either order. 20 squared is equal to 400, and 50
squared is equal to 2500. Adding these gives us a value of 𝑥
squared equal to 2900. Our final step is to square root
both sides of this equation. 𝑥 is equal to the square root of
2900. Typing this into the calculator
gives us 53.851648 and so on.

We are asked to give our answer to
the nearest foot. So, we need to round to the nearest
whole number. As the number in the tens column is
greater than or equal to five, we round up. The length of the guy wire is 54
foot, to the nearest foot. This is a sensible answer as it is
greater than 50 but less than 70. The hypotenuse must be greater than
the other two lengths but smaller than the sum of the shorter two lengths.

We will now look at a second
question that looks at solving a real-life problem.

The figure shows a 129-meter long
bridge on supports 𝑀𝐶 and 𝑀𝐷 attached at the midpoint 𝑀. If 𝐴𝐶 is equal to 51.6 meters,
find the length of 𝑀𝐶 to the nearest hundredth.

We are told in the question that
the length of the bridge 𝐴𝐵 is 129 meters. As 𝑀 is the midpoint of 𝐴𝐵, we
can calculate the distance 𝐴𝑀 by dividing 129 by two. This is equal to 64.5 meters. We are also told that the length
𝐴𝐶 is 51.6 meters. 𝐴𝑀𝐶 is a right triangle where we
know two lengths and need to calculate the length 𝑀𝐶.

We can do this using the
Pythagorean theorem. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared. 𝑐 is the longest side of the right
triangle, known as the hypotenuse. In this case, this is the length
𝑀𝐶. Substituting in our values gives us
64.5 squared plus 51.6 squared is equal to 𝑥 squared. Typing the left-hand side into our
calculator gives us 6822.81. We can then square root both sides
of this equation to calculate the value of 𝑥. 𝑥 is equal to ~~62.600302~~
[82.600302] and on so.

We are asked to round to the
nearest hundredth, which is the same as rounding to two decimal places. As this rounds down, the length of
𝑀𝐶, to the nearest hundredth, is 82.60 meters.

We will now look at a couple of
questions where we use the Pythagorean theorem to solve some geometric problems.

Find the area of the square
𝐵𝐸𝐷𝐶.

As 𝐵𝐸𝐷𝐶 is a square, we know
that each of the lengths is the same size. The area of any square can be
calculated by squaring the length of one of the sides. We also notice that we have a right
triangle where two of the lengths are given. The third length is the length
𝑥. We can, therefore, calculate the
missing length by using the Pythagorean theorem. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is the longest side or hypotenuse.

Substituting in our values, we have
𝑥 squared plus 21 squared is equal to 35 squared. This is because 35 is the
hypotenuse. 21 squared is equal to 441. And 35 squared is 1225. We can subtract 441 from both
sides, giving us 𝑥 squared is equal to 784. Square rooting both sides of this
equation gives us 𝑥 is equal to 28. The length of each side on the
square is 28 centimeters.

In this question, there was
actually a shortcut we could have used to calculate the length of 𝐵𝐶. One of our Pythagorean triples is
three four five. This means that any triangle with
sides in this ratio will be a right triangle. The hypotenuse, or longest side of
our triangle, was 35 centimeters. And one of the shorter sides was 21
centimeters. Three multiplied by seven is 21,
and five multiplied by seven is 35. As four multiplied by seven is
equal to 28, the missing length in the triangle was 28 centimeters. This confirms our previous
calculation.

We can then calculate the area of
the square by squaring 28. As 28 squared is equal to 784, the
area of square 𝐵𝐸𝐷𝐶 is 784 square centimeters. Our answer for area will always be
in square units.

We will now look at a second
geometric problem.

Find the perimeter of 𝐴𝐵𝐶𝐷.

The perimeter of any shape is the
distance around the outside. In this case, we would need to add
the lengths 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, and 𝐷𝐴. Three of these lengths are given to
us. And we will call the length 𝐷𝐴 𝑥
centimeters. Substituting in the values we know
gives us a perimeter of 20 plus 48 plus 39 plus 𝑥. This simplifies to 107 plus 𝑥.

We notice that our quadrilateral or
four-sided shape is split into two right triangles. This means that we could use the
Pythagorean theorem to calculate any missing lengths. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is the longest side or hypotenuse. In this question, however, there is
a quicker method using our knowledge of Pythagorean triples.

Two of these such triples are five
12 13 and three four five. This means that any triangle with
three lengths in these ratios will be a right triangle. Let’s begin by considering the
orange triangle with lengths 20 centimeters and 48 centimeters and hypotenuse
𝑦. Five multiplied by four is equal to
20, and 12 multiplied by four is equal to 48. This means that we can calculate
the length 𝑦 by multiplying 13 by four. This is equal to 52. Therefore, the length of 𝐴𝐶 is 52
centimeters.

In our pink triangle, the two
shorter sides have lengths 39 and 52 centimeters. The length of the hypotenuse, or
longest side, is 𝑥. Multiplying three and four by 13
gives us 39 and 52, respectively. This means that the longest side 𝑥
will be equal to five multiplied by 13. This is equal to 65. The length of 𝑥 or 𝐴𝐷 is 65
centimeters. Substituting this into our
perimeter expression gives us 107 plus 65. 107 plus 65 is equal to 172. We can, therefore, conclude that
the perimeter of 𝐴𝐵𝐶𝐷 is 172 centimeters.

Our final question involves
applying the converse of the Pythagorean theorem.

The distances between three cities
are 77 miles, 36 miles, and 49 miles. Do the positions of these cities
form a right triangle?

We can answer this question by
considering the Pythagorean theorem. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is the longest side or hypotenuse of a
right triangle. The converse of the Pythagorean
theorem states that if the square of the longest side of a triangle is equal to the
sum of the squares of the other two sides, then the triangle is right angled.

In this question, we need to
consider the sum of the squares of 36 and 49 and see if they’re equal to the square
of 77. 77 squared is equal to 5929. 36 squared plus 49 squared is equal
to 3697. These two values are not equal. This means that 36 squared plus 49
squared is not equal to 77 squared. We can, therefore, conclude that as
the three distances do not satisfy the Pythagorean theorem, the triangle is not a
right triangle.

We will now summarize the key
points from this video. The Pythagorean theorem states that
in any right triangle, the square of the hypotenuse is equal to the sum of the
squares of the two shorter sides. This is usually written 𝑎 squared
plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the
hypotenuse. We can apply this theorem to solve
geometric and real-life problems. This includes calculating the
length of the hypotenuse or one of the shorter sides.

Our knowledge of Pythagorean
triples often provides a shortcut. Examples of Pythagorean triples are
three four five and five 12 13. Any triangle with three lengths in
these ratios will be a right triangle. We also know that the converse of
the Pythagorean theorem is true. If the three lengths of any
triangle satisfy 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, then the
triangle is a right triangle.