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Lesson Explainer: Pythagorean Inequality Theorem Mathematics • 8th Grade

In this explainer, we will learn how to determine whether an angle in a triangle is acute, right, or obtuse by using the Pythagorean inequality.

Before we begin discussing the Pythagorean inequality, it is worth recalling the Pythagorean theorem and a property of this theorem.

Theorem: Pythagorean Theorem and Its Application

If 𝐴𝐡𝐢 is a right triangle at 𝐡, then 𝐴𝐢=𝐴𝐡+𝐡𝐢.

The same result is true in reverse and is called the converse of the Pythagorean theorem. This states that if we have a triangle 𝐴𝐡𝐢, where 𝐴𝐢=𝐴𝐡+𝐡𝐢, then we can conclude that the angle at 𝐡 is a right angle.

We can use this result to prove whether a triangle is a right triangle just from its side lengths. For example, if we have a triangle with side lengths 6, 8, and 10, then we note that 10 is the longest side and we have that 10=100 and 8+6=100; so the triangle is a right triangle.

Similarly, if we have a triangle with side lengths of 2, 3, and 4, then we can note that 4=16 and 2+3=13. Thus, 4 is not equal to 2+3 and so the triangle is not a right triangle.

We can extend this result even further by first recalling the concept of an acute triangle and an obtuse triangle.

Definition: Acute and Obtuse Triangles

If all of the internal angles in a triangle are acute angles, then we call it an acute triangle.

If a triangle has an internal obtuse angle, then we call it an obtuse triangle.

We can now determine whether a triangle is acute, right, or obtuse by extending the Pythagorean theorem to the so-called Pythagorean inequality theorem.

Theorem: Pythagorean Inequality Theorem

Let 𝐴𝐡𝐢 be a triangle with the longest side opposite 𝐡.

  • If the square of the longest side is greater than the sum of the squares of the two shorter sides, then the triangle is obtuse at 𝐡.
  • If the square of the longest side is less than the sum of the squares of the two shorter sides, then the triangle is acute.
  • If the square of the longest side is equal to the sum of the squares of the two shorter sides, then the triangle is right angled at 𝐡.

We can also write this as follows:

  • If 𝐴𝐢>𝐴𝐡+𝐡𝐢, then 𝐡 is an obtuse angle.
  • If 𝐴𝐢<𝐴𝐡+𝐡𝐢, then 𝐡 is an acute angle.
  • If 𝐴𝐢=𝐴𝐡+𝐡𝐢, then 𝐡 is a right angle.

To explain why this result holds true, let’s consider two fixed length sides connected by a hinge at a point 𝐡. In other words, we will keep 𝐴 and 𝐡 fixed, but we will rotate point 𝐢 to construct different triangles.

We can rotate these fixed length sides into different legs of a triangle. We are interested in what happens to the side lengths 𝐴𝐢, 𝐴𝐢, and 𝐴𝐢.

If we rotate the sides to make the angle at 𝐡 a right angle, then we have a right triangle, so we know that 𝐴𝐢=𝐴𝐡+𝐡𝐢.

Increasing the measure of angle 𝐡 will make 𝐴𝐢 longer and decreasing the measure of angle 𝐡 will make 𝐴𝐢 shorter. Hence, if angle 𝐡 is obtuse, 𝐴𝐢>𝐴𝐡+𝐡𝐢, and if angle 𝐡 is acute, 𝐴𝐢<𝐴𝐡+𝐡𝐢.

Let’s now see some examples of applying the Pythagorean inequality theorem to determine the type of angle in a triangle given inequalities involving its side lengths.

Example 1: Identifying the Type of an Angle in a Triangle Using the Triangle Inequality Theorem

In triangle 𝐴𝐡𝐢, (𝐴𝐡)+(𝐡𝐢)<(𝐴𝐢). What type of angle is 𝐡?

Answer

To apply the Pythagorean inequality, we want to compare the square of a side length to the sum of the squares of the other two side lengths. We can do this by rearranging the inequality; we note that saying that π‘₯<𝑦 is the same as saying that 𝑦>π‘₯, so (𝐴𝐢)>(𝐴𝐡)+(𝐡𝐢).

The Pythagorean inequality then tells us that if 𝐴𝐡𝐢 is a triangle where (𝐴𝐢)>(𝐴𝐡)+(𝐡𝐢), then 𝐡 is an obtuse angle. Hence, 𝐡 is an obtuse angle.

Example 2: Using the Pythagorean Inequality to Classify an Angle

In triangle π‘‹π‘Œπ‘, (π‘Œπ‘)>(𝑋𝑍)βˆ’(π‘‹π‘Œ). What type of angle is π‘Œ?

Answer

To apply the Pythagorean inequality, we want to compare the square of a side length to the sum of the squares of the other two side lengths. We can do this by rearranging the inequality. Since we have been asked about the angle at π‘Œ, we want to compare the size of the square of the length opposite vertex π‘Œ to the sum of the squares of the lengths of the other two sides.

The side opposite vertex π‘Œ is 𝑋𝑍 so we want to isolate the square of the length of this side in the inequality. We add (π‘‹π‘Œ) to both sides of the inequality to get (π‘Œπ‘)+(π‘‹π‘Œ)>(𝑋𝑍).

Then, we rewrite the inequality as (𝑋𝑍)<(π‘‹π‘Œ)+(π‘Œπ‘).

The Pythagorean inequality gives us information about the measure of the angle at this shared vertex. In particular, since (𝑋𝑍) is smaller than (π‘‹π‘Œ)+(π‘Œπ‘), it tells us that π‘Œ is an acute angle.

If we wanted to determine the type of a triangle from its side lengths, then we could check every angle in a triangle using the Pythagorean inequality theorem. However, this is not necessary if we recall the following property.

Property: Largest Angle in a Triangle

The angle with the largest measure in a triangle is always opposite the longest side.

This allows us to check whether the largest angle in the triangle is acute, obtuse, or right and hence determine the triangle type.

In our next example, we will use this property to determine the angle with the greatest measure in a triangle with given side lengths and then apply the Pythagorean inequality theorem to determine the type of this triangle.

Example 3: Finding the Largest Angle in a Triangle from Its Side Lengths

Triangle 𝐴𝐡𝐢 has side lengths 𝐴𝐡=7cm, 𝐡𝐢=9cm, and 𝐴𝐢=10cm.

  1. Determine the angle with the greatest measure in △𝐴𝐡𝐢.
  2. Determine the type of △𝐴𝐡𝐢 in terms of its angles.

Answer

Part 1

We start by recalling that the angle with the greatest measure in a triangle is always opposite the longest side. Hence, the angle at 𝐡 has the largest measure.

Part 2

We can apply the Pythagorean inequality theorem to determine the type of angle at 𝐡. To do this, we recall that the theorem states the following:

  • If (𝐴𝐢)>(𝐴𝐡)+(𝐡𝐢), then 𝐡 is an obtuse angle.
  • If (𝐴𝐢)<(𝐴𝐡)+(𝐡𝐢), then 𝐡 is an acute angle.
  • If (𝐴𝐢)=(𝐴𝐡)+(𝐡𝐢), then 𝐡 is a right angle.

We have (𝐴𝐢)=10=100, and (𝐴𝐡)+(𝐡𝐢)=7+9=130.

Hence, (𝐴𝐢)<(𝐴𝐡)+(𝐡𝐢) and the angle at 𝐡 is acute. We know that 𝐡 is the angle with the largest measure in the triangle, so all of the angles in the triangle are acute.

Therefore, △𝐴𝐡𝐢 is an acute triangle.

In our next example, we will determine the type of a triangle given its side lengths.

Example 4: Using the Pythagorean Inequality and Given Side Lengths to Classify an Angle

Consider △𝐴𝐡𝐢, with 𝐴𝐡=9, 𝐡𝐢=10, and 𝐴𝐢=11. What kind of triangle is this, in terms of its angles?

Answer

We first note that the largest angle will be opposite the largest side, so angles 𝐴 and 𝐢 must be smaller than 𝐡. Hence, we only need to determine the type of the angle at 𝐡 to determine the type of triangle △𝐴𝐡𝐢.

We then recall that we can determine the type of triangle in terms of its angles by using the Pythagorean inequality theorem which states the following:

  • If (𝐴𝐢)>(𝐴𝐡)+(𝐡𝐢), then 𝐡 is an obtuse angle.
  • If (𝐴𝐢)<(𝐴𝐡)+(𝐡𝐢), then 𝐡 is an acute angle.
  • If (𝐴𝐢)=(𝐴𝐡)+(𝐡𝐢), then 𝐡 is a right angle.

We note that (𝐴𝐢)=11=121,(𝐴𝐡)+(𝐡𝐢)=9+10=81+100=181.

So, (𝐴𝐢)<(𝐴𝐡)+(𝐡𝐢).

This means that the angle at 𝐡 is an acute angle. Hence, 𝐴𝐡𝐢 is an acute triangle.

In our next example, we will determine the type of a triangle by using the lengths of a similar triangle.

Example 5: Using the Pythagorean Inequality and Given Side Lengths to Classify a Triangle

If 𝐴𝐡𝐢 is a triangle whose side lengths are 11 cm, 26.4 cm, and 28.6 cm and it is similar to a triangle π‘‹π‘Œπ‘, determine the type of β–³π‘‹π‘Œπ‘ in terms of its angles.

Answer

We start by recalling that two triangles are similar if their corresponding angles are equal, so we can determine the type of triangle π‘‹π‘Œπ‘ by finding the type of triangle 𝐴𝐡𝐢.

Taking 𝐴𝐡=11cm, 𝐡𝐢=26.4cm, and 𝐴𝐢=28.6cm, we recall that the Pythagorean inequality theorem tells us that for triangle 𝐴𝐡𝐢, the following holds:

  • If (𝐴𝐢)>(𝐴𝐡)+(𝐡𝐢), then 𝐡 is an obtuse angle.
  • If (𝐴𝐢)<(𝐴𝐡)+(𝐡𝐢), then 𝐡 is an acute angle.
  • If (𝐴𝐢)=(𝐴𝐡)+(𝐡𝐢), then 𝐡 is a right angle.

Since 𝐴𝐢 is the longest side, angle 𝐡 is the largest angle in △𝐴𝐡𝐢; identifying its type will therefore allow us to identify the type of △𝐴𝐡𝐢 in terms of its angles.

We have (𝐴𝐢)=28.6=817.96,(𝐴𝐡)+(𝐡𝐢)=11+26.4=817.96.

Since these are equal, we can conclude that the angle at 𝐡 is a right angle and that 𝐴𝐡𝐢 is a right triangle. Finally, since triangle π‘‹π‘Œπ‘ is similar to △𝐴𝐡𝐢, we must also have that it is a right triangle.

Let’s now see an example where we must apply the Pythagorean inequality theorem along with geometric results to determine the type of a triangle.

Example 6: Determining Whether a Triangle is Obtuse or Acute Based on a Diagram

Determine the type of △𝐡𝐢𝐷 in terms of its angles.

Answer

We start by recalling that we can determine whether an angle in a triangle of known lengths is acute or obtuse by using the Pythagorean inequality theorem. We cannot apply this directly since we do not know 𝐡𝐷. To find 𝐡𝐷, we first want to determine the type of triangle 𝐴𝐡𝐢; we can do this by noting that 135=18225,81+108=18225.

Since these are equal, we must have that 𝐴𝐡𝐢 is a right triangle, where the right angle is at 𝐡 since this is opposite the hypotenuse.

We can then note that 𝐡𝐷 is a line from the vertex 𝐡 which bisects the opposite side. In other words, 𝐡𝐷 is a median of the triangle. We can then recall that a median of a right triangle at the right angle will always have length equal to half the hypotenuse, so 𝐡𝐷=12Γ—135=67.5.cm

We then see that 𝐢𝐷 is half of 𝐴𝐢, so 𝐢𝐷=12Γ—135=67.5.cm

This gives us the following.

We can now determine the type of each angle using the Pythagorean inequality identity. However, we can simplify this process slightly by noting that ∠𝐴𝐢𝐡 is an angle in a right triangle, so it is acute and ∠𝐢𝐡𝐷 is smaller than ∠𝐴𝐡𝐢, which is a right angle, so it is also acute. Alternatively, we could note that triangle 𝐢𝐷𝐡 is an isosceles triangle so the angles at 𝐢 and 𝐡 are the same.

Finally, we can determine the type of the angle at 𝐷 by comparing the square of the side length opposite to 𝐷 to the sum of the squares of the two other sides. We have 81=6561,67.5+67.5=9112.5.

Thus, (𝐢𝐡)<(𝐢𝐷)+(𝐡𝐷).

Hence, △𝐡𝐢𝐷 is an acute triangle.

Example 7: Determining a Triangle Type in a Parallelogram

𝐴𝐡𝐢𝐷 is a parallelogram. If 𝐴𝐢=13cm, 𝐴𝐷=13cm, and 𝐷𝐢=5cm, what is the type of △𝐴𝐷𝐢?

Answer

Let’s start by sketching the information we are given. We recall that a parallelogram has parallel opposite sides. We have the following.

We see that △𝐴𝐷𝐢 is an isosceles triangle.

We now recall that the angle with the greatest measure will be opposite the longest side, so the angles at 𝐢 and 𝐷 will be the largest.

We can check the relative size of the angles by using the Pythagorean inequality theorem which states the following:

  • If (𝐴𝐢)>(𝐴𝐷)+(𝐷𝐢), then ∠𝐢𝐷𝐴 is an obtuse angle.
  • If (𝐴𝐢)<(𝐴𝐷)+(𝐷𝐢), then ∠𝐢𝐷𝐴 is an acute angle.
  • If (𝐴𝐢)=(𝐴𝐷)+(𝐷𝐢), then ∠𝐢𝐷𝐴 is a right angle.

In our triangle, we have (𝐴𝐢)=13=169,(𝐴𝐷)+(𝐷𝐢)=13+5=194.

Hence, (𝐴𝐢)<(𝐴𝐷)+(𝐷𝐢), and so ∠𝐢𝐷𝐴 is an acute angle.

Therefore, △𝐴𝐷𝐢 is an acute triangle.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • If all of the internal angles in a triangle are acute angles, then we call it an acute triangle.
  • If a triangle has an internal obtuse angle, then we call it an obtuse triangle.
  • The largest angle in a triangle is opposite its longest side. This means that we can check the type of triangle by only checking the type of the angle opposite the longest side.
  • Let 𝐴𝐡𝐢 be a triangle with the longest side opposite 𝐡.
    • If the square of the longest side is greater than the sum of the squares of the two shorter sides, then the triangle is obtuse at 𝐡.
    • If the square of the longest side is less than the sum of the squares of the two shorter sides, then the triangle is acute.
    • If the square of the longest side is equal to the sum of the squares of the two shorter sides, then the triangle is right angled at 𝐡.

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