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The Pythagorean theorem states that in a right triangle, the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs.
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Does this mean that a triangle where π squared is equal to π squared plus π squared is necessarily a right triangle?
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To answer this question, weβre gonna go through a list of questions to help us decide.
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So let us assume that triangle π΄π΅πΆ is of side lengths π, π, and π, with π squared equal to π squared plus π squared.
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Let triangle π·π΅πΆ be a right triangle of side lengths π, π, and π.
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Triangle π΄π΅πΆ is found here.
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And triangle π·π΅πΆ is found here.
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So what do we know about each of these triangles?
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We know for triangle π΄π΅πΆ, π squared is equal to π squared plus π squared.
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And we know for triangle π·π΅πΆ that itβs a right triangle.
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So we essentially want to know, is triangle π΄π΅πΆ a right triangle?
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So letβs begin with this first question.
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Using the Pythagorean theorem, what can you say about the relationship between π, π, and π?
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So we are talking about triangle π·π΅πΆ.
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So we wanna know what can we say about the side lengths π, π, and π.
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Well, since itβs a right triangle, we can use the Pythagorean theorem.
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And itβs said to use that.
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And for the Pythagorean theorem, using the Pythagorean theorem, we know the square of the longest side is equal to the sum of the squares of the shorter sides.
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So which side is the longest side?
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That would be π, the hypotenuse.
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So π squared is equal to the shorter sides squared.
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And then adding them together, so π squared plus π squared.
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So we could say π squared is equal to π squared plus π squared.
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Now our next question says, we know that for triangle π΄π΅πΆ, π squared is equal to π squared plus π squared.
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What do you conclude about π?
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So we are told that π squared is equal to π squared plus π squared.
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But we know something else thatβs equal to π squared plus π squared.
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Thatβs π squared.
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This means π squared and π squared must be equal.
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And if π squared is equal to π squared, we could square root both sides and say that π is equal to π.
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The next question says, is it possible to construct different triangles with the same length sides?
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So letβs take this triangle.
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If we were to take the pieces apart and reconstruct another triangle, will it look the same as the old one?
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It may be flipped upside down.
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But essentially, itβs still gonna be the same triangle.
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So is it possible to-to construct different triangles with the same length sides?
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The answer is no.
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So what do you conclude about triangle π΄π΅πΆ?
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Is it necessarily a right triangle?
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The fact that we know, π is equal to π, we can replace π with π.
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Then our side lengths would be exactly the same.
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And we concluded itβs not possible to construct different triangles with the same length sides.
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So if triangle π·π΅πΆ is a right triangle, so is triangle π΄π΅πΆ.
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Therefore, it is congruent to triangle π·π΅πΆ.
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So it is a right triangle at π.