Pythagorean Theorem

1. Diagram
   Observation

   On the general right-triangle sketch at the upper right, one leg forming the right angle is labeled a.

2. Audio
   Observation

   The instructor says that a squared means this leg squared while writing $a^2$ in the formula.

a

One of the two legs of a right triangle, i.e. a side adjacent to the right angle.

Positive real length; interchangeable with b as either leg.

1. Diagram
   Observation

   On the same general right-triangle sketch, the other leg forming the right angle is labeled b.

2. Audio
   Observation

   The instructor says plus b squared, this leg squared, while writing + $b^2$.

b

The other leg of the right triangle, the second side adjacent to the right angle.

Positive real length; interchangeable with a as either leg.

1. Diagram
   Observation

   The side opposite the right-angle square marker on the general triangle is labeled c.

2. Audio
   Observation

   The instructor identifies the side across from the right angle as the hypotenuse and then writes = $c^2$.

c

The hypotenuse of the right triangle, the side opposite the right angle.

Positive real length; in a right triangle it is the longest side.

1. Diagram
   Observation

   In example ①, the side opposite the right angle is labeled x.

2. Formula
   Observation

   The worked equation uses $x^2$ on the right-hand side: $3^2 + 4^2 = x^2$.

3. Audio
   Observation

   The instructor says the hypotenuse in this case we just called x, so x squared.

x

Unknown hypotenuse length in example ①.

Positive real number; solved value shown as $x = 5$.

1. Diagram
   Observation

   In example ②, one leg adjacent to the right angle is labeled x.

2. Formula
   Observation

   The setup written near the end is $x^2 + 12^2 = 13^2$.

3. Audio
   Observation

   The instructor introduces another example where you are solving for the leg now.

1. The final numerical value of x is not reached within this clip.

x

Unknown leg length in example ②.

Positive real number; only the setup equation is completed in this clip.

1. Formula
   Observation

   The board shows $a^2 + b^2 = c^2$ under the general triangle.

2. Audio
   Observation

   The instructor states the Pythagorean theorem as a squared plus b squared equals c squared.

$a^2 + b^2 = c^2$

Algebraic relation among the two legs and the hypotenuse of a right triangle.

Applies to right triangles with side lengths a, b, c where c is opposite the right angle.

1. Formula
   Observation

   The variable x is written on the vertical leg of the second right triangle and in the equation $x^2 + 12^2 = 13^2$.

2. Audio
   Observation

   The speaker refers to x as 'this missing leg'.

x

The unknown length of the missing leg of the right triangle.

Positive real numbers (length).

1. Formula
   Observation

   The variable a is written on the vertical leg of the first right triangle with $a=3$, and in the general formula $a^2 + b^2 = c^2$.

a

The length of one leg of a right triangle.

Positive real numbers (length).

1. Formula
   Observation

   The variable b is written on the horizontal leg of the first right triangle with $b=4$, and in the general formula $a^2 + b^2 = c^2$.

b

The length of the other leg of a right triangle.

Positive real numbers (length).

1. Formula
   Observation

   The variable c is written on the hypotenuse of both right triangles, and in the general formula $a^2 + b^2 = c^2$.

c

The length of the hypotenuse of a right triangle.

Positive real numbers (length).

1. Diagram
   Observation

   A right triangle is drawn with a square marker at the right angle; the two sides meeting there are labeled a and b, and the opposite side is labeled c.

2. Audio
   Observation

   The instructor says the two sides that make up the right angle are called the legs or the sides, and the side across from the right angle, the longest side, is called the hypotenuse.

In a right triangle, the two sides that form the right angle are the legs, and the side opposite the right angle is the hypotenuse. The video explicitly identifies the hypotenuse as the longest side.

1. The triangle must be a right triangle.

2. The hypotenuse is specifically the side opposite the right angle, not merely any long side.

1. Formula
   Observation

   The board displays $a^2 + b^2 = c^2$ beneath the general triangle.

2. Audio
   Observation

   The instructor says the Pythagorean theorem tells us that a squared plus b squared is equal to c squared, the hypotenuse squared.

For a right triangle with legs a and b and hypotenuse c, the sum of the squares of the legs equals the square of the hypotenuse.

1. The triangle is a right triangle.

2. a and b denote the legs.

3. c denotes the hypotenuse, the side opposite the right angle.

1. Legs and hypotenuse of a right triangle

1. Animation
   Observation

   Squares are constructed outward on sides a, b, and c of the general triangle.

2. Audio
   Observation

   The instructor explains geometrically that if you make each side into a square and find its area, the two smaller areas added together equal the area of the third square, which is the hypotenuse squared.

The theorem can be visualized by building squares on the three sides of the right triangle: the combined area of the squares on the two legs equals the area of the square on the hypotenuse.

1. Interpretation applies to the same right triangle setting as the theorem.

2. Each square has side length equal to the corresponding triangle side.

1. Pythagorean theorem statement

1. Formula
   Observation

   Example ① is labeled with $a = 3$, $b = 4$, $c = x$, then $3^2 + 4^2 = x^2$, $9 + 16 = x^2$, $25 = x^2$, sqrt(25) = sqrt($x^2$), and $x = 5$.

2. Audio
   Observation

   The instructor says to substitute into the Pythagorean theorem, simplify, and take the square root of both sides to get x by itself.

When the two legs are known and the hypotenuse is unknown, assign the legs to a and b, assign the unknown hypotenuse to c, substitute into $a^2 + b^2 = c^2$, simplify, and take the positive square root to solve for the side length.

1. The triangle is a right triangle.

2. The unknown is the side opposite the right angle.

3. Side lengths are taken as positive.

1. Pythagorean theorem statement
2. Legs and hypotenuse of a right triangle

1. Diagram
   Observation

   Example ② shows a right triangle with one leg labeled x, the other leg labeled 12, and the hypotenuse labeled 13.

2. Formula
   Observation

   The instructor writes $x^2 + 12^2 = 13^2$.

3. Audio
   Observation

   He says this is another example where you are solving for the leg now and warns to get a, b, and c correctly.

1. The clip ends before the algebraic solution for x is completed.

When one leg is unknown, still place the unknown leg with the other leg on the left side of the equation and put the hypotenuse on the right side. In the second example this gives $x^2 + 12^2 = 13^2$.

1. The triangle is a right triangle.

2. The unknown x is a leg, not the hypotenuse.

3. 13 is the side opposite the right angle.

1. Pythagorean theorem statement
2. Legs and hypotenuse of a right triangle

1. Formula
   Observation

   The general formula $a^2 + b^2 = c^2$ is written on the whiteboard next to a diagram of squares built on the sides of a right triangle.

2. Audio
   Observation

   The speaker applies the theorem by setting up the equation $x^2 + 12^2 = 13^2$ for a right triangle with legs x and 12, and hypotenuse 13.

The Pythagorean theorem 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 (the legs).

1. The triangle must be a right-angled triangle.

2. a and b represent the lengths of the legs.

3. c represents the length of the hypotenuse.

1. Audio
   Observation

   The speaker says, 'now we just want to get x squared by itself. So let's subtract 144 from both sides of the equation.'

2. Formula
   Observation

   The steps show subtracting 144 from both sides to get $x^2 = 25$, and then taking the square root of both sides to get $x = 5$.

To isolate a variable in an equation, perform inverse operations on both sides. Here, subtraction is used to isolate the squared term, and taking the square root is used to solve for the variable.

1. Operations must be applied equally to both sides of the equation to maintain equality.

1. Pythagorean Theorem

1. Formula
   Observation

   The board shows $a^2 + b^2 = c^2$.

2. Audio
   Observation

   The instructor states the theorem verbally as a squared plus b squared equals c squared.

If a and b are the legs of a right triangle and c is the hypotenuse, then $a^2 + b^2 = c^2$.

1. The figure is a right triangle.

2. a and b are the sides forming the right angle.

3. c is the side opposite the right angle.

For every right triangle with legs a, b and hypotenuse c.

1. Audio
   Observation

   The instructor calls the side across from the right angle that longest side the hypotenuse.

2. Diagram
   Observation

   The side labeled c is visually the side opposite the right-angle marker.

In the video's explanation, the hypotenuse is identified as the longest side of the right triangle.

1. The triangle is a right triangle.

2. The side being discussed is opposite the right angle.

For the right triangles shown and discussed in the clip.

1. Formula
   Observation

   The board successively shows $3^2 + 4^2 = x^2$, then $9 + 16 = x^2$, then $25 = x^2$, then sqrt(25) = sqrt($x^2$), then $x = 5$.

2. Audio
   Observation

   The instructor narrates substitution, simplification, addition, and taking the square root of both sides.

1. Expression

   $$a = 3,\ b = 4,\ c = x$$
   Explanation

   Identify the known legs and the unknown hypotenuse in the first triangle.

   Justification

   The side labels on the diagram match the roles in the theorem: the two sides forming the right angle are 3 and 4, and the opposite side is x.

   Shown in the video
2. Expression

   $$3^2 + 4^2 = x^2$$
   Explanation

   Substitute the side lengths into the Pythagorean theorem.

   Justification

   This follows directly from $a^2 + b^2 = c^2$ after assigning $a = 3$, $b = 4$, $c = x$.

   Shown in the video
3. Expression

   $$9 + 16 = x^2$$
   Explanation

   Evaluate the squares on the left-hand side.

   Justification

   Arithmetic simplification: $3^2 = 9$ and $4^2 = 16$.

   Shown in the video
4. Expression

   $$25 = x^2$$
   Explanation

   Add the two numbers on the left-hand side.

   Justification

   Arithmetic simplification: $9 + 16 = 25$.

   Shown in the video
5. Expression

   $$\sqrt{25} = \sqrt{x^2}$$
   Explanation

   Take the square root of both sides to isolate x.

   Justification

   The instructor explicitly says to take the square root of both sides to get x by itself.

   Shown in the video
6. Expression

   $$x = 5$$
   Explanation

   Conclude the hypotenuse length.

   Justification

   Since $25 = 5^2$ and the context is a triangle side length, the positive root is used.

   Shown in the video

The unknown hypotenuse in example ① is $x = 5$.

1. Formula
   Observation

   The board shows $x^2 + 12^2 = 13^2$ for the second triangle.

2. Audio
   Observation

   The instructor says remember that a and b are the two sides that make up the right angle and that the one across from the right angle is your c side.

1. No further algebraic steps are shown before the clip ends.

1. Expression

   $$\text{legs: } x,\ 12;\ \text{hypotenuse: } 13$$
   Explanation

   Identify which sides are legs and which side is the hypotenuse in the second triangle.

   Justification

   The right-angle marker is between the sides labeled x and 12, while 13 lies opposite that marker.

   Shown in the video
2. Expression

   $$x^2 + 12^2 = 13^2$$
   Explanation

   Substitute these labels into the Pythagorean theorem.

   Justification

   Using $a^2 + b^2 = c^2$ with a and b as the legs and c as the hypotenuse gives this equation.

   Shown in the video

The correct equation for the second example is $x^2 + 12^2 = 13^2$; the solution for x is not completed in this clip.

1. Formula
   Observation

   The step-by-step algebraic manipulation is written on the whiteboard: $x^2 + 12^2 = 13^2$, $x^2 + 144 = 169$, $x^2 = 25$, $\sqrt{x^2} = \sqrt{25}$, $x = 5$.

2. Audio
   Observation

   The speaker narrates each step of the calculation, including squaring the numbers, subtracting 144 from both sides, and taking the square root.

1. Expression

   $$x^2 + 12^2 = 13^2$$
   Explanation

   Set up the Pythagorean theorem equation using the given side lengths of the right triangle.

   Justification

   Pythagorean theorem definition.

   Shown in the video
2. Expression

   $$x^2 + 144 = 169$$
   Explanation

   Calculate the squares of the known side lengths.

   Justification

   Arithmetic evaluation.

   Shown in the video
3. Expression

   $$x^2 = 25$$
   Explanation

   Subtract 144 from both sides of the equation to isolate $x^2$.

   Justification

   Algebraic manipulation (subtraction property of equality).

   Shown in the video
4. Expression

   $$\sqrt{x^2} = \sqrt{25}$$
   Explanation

   Take the square root of both sides to solve for x.

   Justification

   Algebraic manipulation (square root property).

   Shown in the video
5. Expression

   $$x = 5$$
   Explanation

   Evaluate the square root of 25.

   Justification

   Arithmetic evaluation.

   Shown in the video

The length of the missing leg x is 5.

1. Diagram
   Observation

   Example ① is a right triangle with legs 3 and 4 and hypotenuse x.

2. Formula
   Observation

   The worked lines on the board are $3^2 + 4^2 = x^2$, $9 + 16 = x^2$, $25 = x^2$, sqrt(25) = sqrt($x^2$), $x = 5$.

3. Audio
   Observation

   The instructor says let me show you how this works with this first example and then narrates the substitution and square-root step.

Given a right triangle with legs 3 and 4, find the hypotenuse x.

1. Right triangle.

2. Leg lengths are 3 and 4.

3. The unknown side x is opposite the right angle.

Determine the length of the hypotenuse x.

1. Expression

   $$a = 3,\ b = 4,\ c = x$$
   Explanation

   Assign the known legs to a and b and the unknown hypotenuse to c.

   Justification

   This matches the definition of legs and hypotenuse used earlier in the clip.

   Shown in the video
2. Expression

   $$3^2 + 4^2 = x^2$$
   Explanation

   Substitute into the Pythagorean theorem.

   Justification

   Direct application of $a^2 + b^2 = c^2$.

   Shown in the video
3. Expression

   $$9 + 16 = x^2$$
   Explanation

   Square the leg lengths.

   Justification

   Basic arithmetic simplification.

   Shown in the video
4. Expression

   $$25 = x^2$$
   Explanation

   Add the squared leg lengths.

   Justification

   $9 + 16 = 25$.

   Shown in the video
5. Expression

   $$\sqrt{25} = \sqrt{x^2}$$
   Explanation

   Take square roots of both sides.

   Justification

   The instructor states this is how to get x by itself.

   Shown in the video
6. Expression

   $$x = 5$$
   Explanation

   Read off the positive solution.

   Justification

   The side length is positive, so the relevant root is 5.

   Shown in the video

$x = 5$

Substitution check: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$, so the result satisfies the Pythagorean theorem.

1. Diagram
   Observation

   Example ② is a right triangle with one leg x, another leg 12, and hypotenuse 13.

2. Formula
   Observation

   The board shows $x^2 + 12^2 = 13^2$.

3. Audio
   Observation

   The instructor says let me show you another example where you're solving for the leg now and warns to get a, b, and c correctly.

1. The clip stops before solving for x.

Given a right triangle with one leg x, another leg 12, and hypotenuse 13, set up the Pythagorean equation for x.

1. Right triangle.

2. One leg is unknown and labeled x.

3. The other leg is 12.

4. The side opposite the right angle is 13.

Write the correct equation using the Pythagorean theorem.

1. Expression

   $$\text{legs: } x,\ 12;\ \text{hypotenuse: } 13$$
   Explanation

   Identify the roles of the sides before substituting.

   Justification

   The instructor emphasizes that a and b are the sides making the right angle and c is the side across from it.

   Shown in the video
2. Expression

   $$x^2 + 12^2 = 13^2$$
   Explanation

   Substitute the side lengths into $a^2 + b^2 = c^2$.

   Justification

   This is direct application of the theorem with the unknown placed among the legs.

   Shown in the video

$x^2 + 12^2 = 13^2$

The setup is consistent with the diagram because 13 is opposite the right-angle marker and therefore must appear on the hypotenuse side of the equation.

1. Formula
   Observation

   A right triangle is drawn with a vertical leg labeled x, a horizontal leg labeled 12, and a hypotenuse labeled 13.

2. Audio
   Observation

   The speaker solves for x, concluding 'that's going to be this missing leg here'.

Find the length of the missing leg (x) of a right triangle with a known leg of 12 and a hypotenuse of 13.

1. Right triangle with legs x and 12, and hypotenuse 13.

Determine the value of x.

1. Expression

   $$x^2 + 12^2 = 13^2$$
   Explanation

   Apply the Pythagorean theorem.

   Justification

   Theorem application.

   Shown in the video
2. Expression

   $$x^2 + 144 = 169$$
   Explanation

   Square the known values.

   Justification

   Arithmetic.

   Shown in the video
3. Expression

   $$x^2 = 25$$
   Explanation

   Subtract 144 from both sides.

   Justification

   Algebraic manipulation.

   Shown in the video
4. Expression

   $$x = 5$$
   Explanation

   Take the square root of both sides.

   Justification

   Algebraic manipulation.

   Shown in the video

$x = 5$

Check if $5^2 + 12^2 = 13^2$: $25 + 144 = 169$, which is true.

1. Diagram
   Observation

   A general right triangle is shown with a square marker at the right angle and labels a, b, c on the three sides.

2. Audio
   Observation

   The instructor points out the two sides making the right angle and the side across from it.

1. Right triangle

2. Right-angle square marker

3. Labels a, b, c

1. The instructor directs attention first to the two sides forming the right angle.

2. Then attention shifts to the side opposite the right angle.

1. The triangle remains a right triangle throughout the labeling.

2. The side opposite the right angle stays identified as c.

The visual establishes the convention used in the formula: a and b are legs, c is the hypotenuse.

1. Animation
   Observation

   Squares are drawn outward on sides a, b, and c of the general triangle.

2. Audio
   Observation

   The instructor describes making each side into a square, finding the areas, and adding the two smaller areas to equal the third square's area.

1. Right triangle

2. Square on side a

3. Square on side b

4. Square on side c

1. A square is added on side a.

2. A square is added on side b.

3. A larger square is added on side c.

1. Each square shares one side with the corresponding triangle side.

2. The relationship being illustrated is between areas of the three squares.

The drawing gives a geometric picture of $a^2 + b^2 = c^2$ as an area identity.

1. Diagram
   Observation

   Example ① shows a right triangle with legs 3 and 4 and hypotenuse x.

2. Formula
   Observation

   The board accumulates the solution lines from $3^2 + 4^2 = x^2$ down to $x = 5$.

1. Triangle ①

2. Equation lines

3. Final circled answer $x = 5$

1. The unknown hypotenuse is relabeled conceptually as $c = x$.

2. Successive algebraic lines are written below the triangle.

3. The final value $x = 5$ is emphasized.

1. The triangle's given leg lengths remain 3 and 4.

2. The unknown remains the side opposite the right angle.

The visual sequence demonstrates substitution into the theorem followed by simplification and square-root extraction.

1. Diagram
   Observation

   Example ② shows a right triangle with leg x, leg 12, and hypotenuse 13.

2. Formula
   Observation

   Only the setup line $x^2 + 12^2 = 13^2$ is written before the clip ends.

1. No solution steps beyond the setup are visible in this clip.

1. Triangle ②

2. Labels x, 12, 13

3. Equation $x^2 + 12^2 = 13^2$

1. Attention shifts from the first solved example to a new triangle.

2. The instructor identifies the side roles again.

3. The equation is set up but not solved on-screen within this clip.

1. 13 remains the side opposite the right angle.

2. x remains one of the legs.

This visual contrasts with example ① by placing the unknown on the leg side of the equation rather than the hypotenuse side.

1. Diagram
   Observation

   The whiteboard displays the title 'PYTHAGOREAN THEOREM', two numbered example problems with right triangles, and a general diagram showing squares on the sides of a right triangle with the formula $a^2 + b^2 = c^2$.

1. Title text

2. Two right triangle examples

3. General Pythagorean theorem diagram

4. Algebraic equations

1. The speaker writes the step-by-step solution for the second example problem.

1. The first example problem and the general diagram remain unchanged on the board.

The visual layout provides context for the specific problem being solved, reinforcing the general theorem and showing a previously solved example.

1. Animation
   Observation

   The video transitions to an end screen with a teal background, a smaller video window of the speaker, and text prompts 'Learn Next', 'Improve with my Courses', and 'Subscribe' with a red arrow.

1. Teal background

2. Smaller video window

3. Text prompts

4. Red arrow

1. The main whiteboard view is replaced by the end screen layout.

1. The speaker continues talking in the smaller video window.

This visual change indicates the end of the instructional content and provides calls to action for the viewer.

1. Audio
   Observation

   The instructor says sometimes this confuses students and stresses that you want to make sure that you get the a, b, and c correctly.

2. Audio
   Observation

   He adds that it does not matter whether you call one leg a or b, but the important thing is that the one across from the right angle is your c side.

Students may assign the unknown side to c incorrectly or think the labels a and b must correspond to fixed positions.

The video clarifies that a and b can be interchanged because they are just the two legs, but c must always be the side opposite the right angle.

1. Audio
   Observation

   The instructor explicitly introduces example ② as solving for the leg now, contrasting it with the previous example.

2. Formula
   Observation

   The setup places $x^2$ with $12^2$ on the left and $13^2$ on the right.

Learners may assume the unknown side is always the hypotenuse or always goes on the right-hand side of the equation.

The second example shows that when the unknown is a leg, it belongs with the other leg on the left side of $a^2 + b^2 = c^2$.

1. Audio
   Observation

   The speaker states, 'Now sometimes when you're doing these problems, you're going to get a square root that you can't simplify and you have to simplify that radical.'

Students might assume that taking the square root in the Pythagorean theorem will always result in a whole number.

Often, the result is an irrational number or a radical that needs to be simplified (e.g., $\sqrt{50} = 5\sqrt{2}$).

1. Audio
   Observation

   The instructor defines legs and hypotenuse immediately before stating $a^2 + b^2 = c^2$.

2. Diagram
   Observation

   The same triangle labeling is used for both the definitions and the formula.

Correct use of the theorem depends on first identifying which sides are the legs and which side is the hypotenuse.

1. Formula
   Observation

   The algebraic identity $a^2 + b^2 = c^2$ is already on the board.

2. Animation
   Observation

   Squares are then drawn on the three sides while the instructor explains the area interpretation.

The area picture is a geometric interpretation of the same theorem written algebraically.

1. Formula
   Observation

   Example ① substitutes 3, 4, and x into $a^2 + b^2 = c^2$.

2. Audio
   Observation

   The instructor says let me show you how this works with this first example.

The first worked example applies the theorem to find an unknown hypotenuse from two known legs.

1. Audio
   Observation

   After finishing example ①, the instructor says let me show you another example where you're solving for the leg now.

2. Formula
   Observation

   Example ② changes the placement of the unknown from the hypotenuse side to a leg side.

The second example contrasts with the first by solving for a leg instead of the hypotenuse, showing that side identification matters more than variable position in the diagram.

1. Audio
   Observation

   The speaker transitions from stating the theorem to performing algebraic operations to isolate the variable.

Solving for a missing side using the Pythagorean theorem requires applying algebraic methods to isolate the variable.

1. Audio
   Observation

   The speaker mentions that sometimes the result of the square root operation requires simplifying a radical.

While the theorem involves a square root, the result is not always a simple integer, contrasting with the misconception that it always is.

1. Audio
   Observation

   The instructor defines the side across from the right angle as the hypotenuse.

2. Diagram
   Observation

   The side labeled c is opposite the right-angle marker.

1. Legs and hypotenuse of a right triangle

1. Formula
   Observation

   The board shows $a^2 + b^2 = c^2$.

2. Audio
   Observation

   The instructor states the theorem verbally.

1. Pythagorean theorem statement

1. Animation
   Observation

   Squares are drawn on all three sides.

2. Audio
   Observation

   The instructor explains the sum of the two smaller square areas equals the larger square area.

1. Geometric area interpretation of the theorem
2. Pythagorean theorem statement

1. Formula
   Observation

   The worked solution ends with $x = 5$.

2. Audio
   Observation

   The instructor says now we know the length of this hypotenuse.

1. Method for finding the hypotenuse when both legs are known
2. Find the hypotenuse from legs 3 and 4

1. Formula
   Observation

   The second example is set up as $x^2 + 12^2 = 13^2$.

2. Audio
   Observation

   The instructor says this is an example where you are solving for the leg now.

1. Setting up the theorem when a leg is unknown
2. Set up an equation when a leg is unknown
3. Confusing which side is c

1. Audio
   Observation

   The instructor says it doesn't matter whether you call this side a or this side a or this side b or this side b, but the one across from the right angle is your c side.

1. Legs and hypotenuse of a right triangle
2. Confusing which side is c

1. Audio
   Observation

   The entire segment demonstrates finding a missing leg.

1. Pythagorean Theorem
2. Solving an Algebraic Equation
3. Finding a Missing Leg
4. Derivation for Example 2

1. Formula
   Observation

   The formula $a^2 + b^2 = c^2$ is prominently displayed.

1. Pythagorean Theorem

1. Audio
   Observation

   The speaker explicitly mentions getting a square root that you can't simplify and having to simplify the radical.

1. Assuming Square Roots Always Yield Integers