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 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. If π΄π΅πΆ is a right triangle at π΅, then π΄πΆ=π΄π΅+π΅πΆο¨ο¨ο¨. The same result is true in reverse and is called the converse of the Pythagorean theorem. This states that if we have a triangle π΄π΅πΆ, where π΄πΆ=π΄π΅+π΅πΆο¨ο¨ο¨, then we can conclude that the angle at π΅ 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=100ο¨ and 8+6=100ο¨ο¨; 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=16ο¨ and 2+3=13ο¨ο¨. Thus, 4ο¨ is not equal to 2+3ο¨ο¨ 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. 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. Let π΄π΅πΆ be a triangle with the longest side opposite π΅. We can also write this as follows: To explain why this result holds true, letβs consider two fixed length sides connected by a hinge at a point π΅. In other words, we will keep π΄ and π΅ fixed, but we will rotate point πΆ 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 π΄πΆο§, π΄πΆο¨, and π΄πΆο©. If we rotate the sides to make the angle at π΅ a right angle, then we have a right triangle, so we know that π΄πΆ=π΄π΅+π΅πΆο¨ο¨ο¨. Increasing the measure of angle π΅ will make π΄πΆ longer and decreasing the measure of angle π΅ will make π΄πΆ shorter. Hence, if angle π΅ is obtuse, π΄πΆ>π΄π΅+π΅πΆο¨ο¨ο¨, and if angle π΅ is acute, π΄πΆ<π΄π΅+π΅πΆο¨ο¨ο¨. Letβs 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. In triangle π΄π΅πΆ, (π΄π΅)+(π΅πΆ)<(π΄πΆ)ο¨ο¨ο¨. What type of angle is π΅? AnswerTo 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₯<π¦ is the same as saying that π¦>π₯, so (π΄πΆ)>(π΄π΅)+(π΅πΆ).ο¨ο¨ο¨ The Pythagorean inequality then tells us that if π΄π΅πΆ is a triangle where (π΄πΆ)>(π΄π΅)+(π΅πΆ)ο¨ο¨ο¨, then π΅ is an obtuse angle. Hence, π΅ is an obtuse angle. In triangle πππ, (ππ)>(ππ)β(ππ)ο¨ο¨ο¨. What type of angle is π? AnswerTo 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, we want to compare the size of the square of the length opposite vertex π to the sum of the squares of the lengths of the other two sides. The side opposite vertex π is ππ so we want to isolate the square of the length of this side in the inequality. We add (ππ)ο¨ to both sides of the inequality to get (ππ)+(ππ)>(ππ).ο¨ο¨ο¨ Then, we rewrite the inequality as (ππ)<(ππ)+(ππ).ο¨ο¨ο¨ The Pythagorean inequality gives us information about the measure of the angle at this shared vertex. In particular, since (ππ)ο¨ is smaller than (ππ)+(ππ)ο¨ο¨, it tells us that π 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. 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. Triangle π΄π΅πΆ has side lengths π΄π΅=7cm, π΅πΆ=9cm, and π΄πΆ=10cm.
AnswerPart 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 π΅ has the largest measure. Part 2 We can apply the Pythagorean inequality theorem to determine the type of angle at π΅. To do this, we recall that the theorem states the following:
We have (π΄πΆ)=10=100,ο¨ο¨ and (π΄π΅)+(π΅πΆ)=7+9=130.ο¨ο¨ο¨ο¨ Hence, (π΄πΆ)<(π΄π΅)+(π΅πΆ)ο¨ο¨ο¨ and the angle at π΅ is acute. We know that π΅ is the angle with the largest measure in the triangle, so all of the angles in the triangle are acute. Therefore, β³π΄π΅πΆ is an acute triangle. In our next example, we will determine the type of a triangle given its side lengths. Consider β³π΄π΅πΆ, with π΄π΅=9, π΅πΆ=10, and π΄πΆ=11. What kind of triangle is this, in terms of its angles? AnswerWe first note that the largest angle will be opposite the largest side, so angles π΄ and πΆ must be smaller than π΅. Hence, we only need to determine the type of the angle at π΅ to determine the type of triangle β³π΄π΅πΆ. 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:
We note that (π΄πΆ)=11=121,(π΄π΅)+(π΅πΆ)=9+10=81+100=181.ο¨ο¨ο¨ο¨ο¨ο¨ So, (π΄πΆ)<(π΄π΅)+(π΅πΆ).ο¨ο¨ο¨ This means that the angle at π΅ is an acute angle. Hence, π΄π΅πΆ is an acute triangle. In our next example, we will determine the type of a triangle by using the lengths of a similar triangle. If π΄π΅πΆ is a triangle whose side lengths are 11 cm, 26.4 cm, and 28.6 cm and it is similar to a triangle πππ, determine the type of β³πππ in terms of its angles. AnswerWe start by recalling that two triangles are similar if their corresponding angles are equal, so we can determine the type of triangle πππ by finding the type of triangle π΄π΅πΆ. Taking π΄π΅=11cm, π΅πΆ=26.4cm, and π΄πΆ=28.6cm, we recall that the Pythagorean inequality theorem tells us that for triangle π΄π΅πΆ, the following holds:
Since π΄πΆ is the longest side, angle π΅ is the largest angle in β³π΄π΅πΆ; identifying its type will therefore allow us to identify the type of β³π΄π΅πΆ in terms of its angles. We have (π΄πΆ)=28.6=817.96,(π΄π΅)+(π΅πΆ)=11+26.4=817.96.ο¨ο¨ο¨ο¨ο¨ο¨ Since these are equal, we can conclude that the angle at π΅ is a right angle and that π΄π΅πΆ is a right triangle. Finally, since triangle πππ is similar to β³π΄π΅πΆ, we must also have that it is a right triangle. Letβs now see an example where we must apply the Pythagorean inequality theorem along with geometric results to determine the type of a triangle. Determine the type of β³π΅πΆπ· in terms of its angles. AnswerWe 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 π΅π·. To find π΅π·, we first want to determine the type of triangle π΄π΅πΆ; we can do this by noting that 135=18225,81+108=18225.ο¨ο¨ο¨ Since these are equal, we must have that π΄π΅πΆ is a right triangle, where the right angle is at π΅ since this is opposite the hypotenuse. We can then note that π΅π· is a line from the vertex π΅ which bisects the opposite side. In other words, π΅π· 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 π΅π·=12Γ135=67.5.cm We then see that πΆπ· is half of π΄πΆ, so πΆπ·=12Γ135=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 β π΄πΆπ΅ is an angle in a right triangle, so it is acute and β πΆπ΅π· is smaller than β π΄π΅πΆ, which is a right angle, so it is also acute. Alternatively, we could note that triangle πΆπ·π΅ is an isosceles triangle so the angles at πΆ and π΅ are the same. Finally, we can determine the type of the angle at π· by comparing the square of the side length opposite to π· to the sum of the squares of the two other sides. We have 81=6561,67.5+67.5=9112.5.ο¨ο¨ο¨ Thus, (πΆπ΅)<(πΆπ·)+(π΅π·).ο¨ο¨ο¨ Hence, β³π΅πΆπ· is an acute triangle. π΄π΅πΆπ· is a parallelogram. If π΄πΆ=13cm, π΄π·=13cm, and π·πΆ=5cm, what is the type of β³π΄π·πΆ? AnswerLetβs start by sketching the information we are given. We recall that a parallelogram has parallel opposite sides. We have the following. We see that β³π΄π·πΆ is an isosceles triangle. We now recall that the angle with the greatest measure will be opposite the longest side, so the angles at πΆ and π· will be the largest. We can check the relative size of the angles by using the Pythagorean inequality theorem which states the following:
In our triangle, we have (π΄πΆ)=13=169,(π΄π·)+(π·πΆ)=13+5=194.ο¨ο¨ο¨ο¨ο¨ο¨ Hence, (π΄πΆ)<(π΄π·)+(π·πΆ)ο¨ο¨ο¨, and so β πΆπ·π΄ is an acute angle. Therefore, β³π΄π·πΆ is an acute triangle. Letβs finish by recapping some of the important points from this explainer.
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What if the square of the longest side is less than the sum of the square of the shorter side?
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