Question Video: Determining Whether a Triangle is Obtuse or Acute or a Right Triangle Using Its Side Lengths | Nagwa Question Video: Determining Whether a Triangle is Obtuse or Acute or a Right Triangle Using Its Side Lengths | Nagwa

# Question Video: Determining Whether a Triangle is Obtuse or Acute or a Right Triangle Using Its Side Lengths Mathematics • Second Year of Preparatory School

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π΄π΅πΆπ· is a parallelogram. If π΄πΆ = 13 cm, π΄π· = 13 cm, and π·πΆ = 5 cm, what is the type of β³π΄π·πΆ?

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### Video Transcript

π΄π΅πΆπ· is a parallelogram. If π΄πΆ equals 13 centimeters, π΄π· equals 13 centimeters, and π·πΆ equals five centimeters, what is the type of triangle π΄π·πΆ?

We see that triangle π΄π·πΆ is an isosceles triangle. And recalling that the angle in a triangle with the greatest measure is opposite the longest side, in triangle π΄π·πΆ the angles at πΆ and π·, which are equal, will have the largest measure. Choosing either one of the angles at πΆ and π·, we can use the Pythagorean inequality theorem to confirm that these angles are acute.

Taking the angle at π· to work on, this theorem tells us three things. First, that if the square of the longest side is greater than the sum of the squares of the other two sides, then the angle opposite the longest side is an obtuse angle. Second, if the square of the longest side is less than the sum of squares of the other two sides, the angle is acute. And third, if the square of the longest side is equal to the sum of the squares of the other two, then the angle opposite is a right angle.

In our case, we have π΄πΆ squared, that is 13 squared, equals 169 and that π΄π· squared plus π·πΆ squared equals 13 squared plus five squared. And thatβs equal to 194. Hence, π΄πΆ squared is less than π΄π· squared plus π·πΆ squared. And so angle πΆπ·π΄ is an acute angle. Angle π΄πΆπ· is the same, so this is also acute. And since these angles have the largest measure in triangle π΄π·πΆ, angle πΆπ΄π· must be smaller than them. Hence, the third angle, angle πΆπ΄π· is also acute. Since all three angles are acute and, in particular, the angle with the largest measure is acute, triangle π΄π·πΆ is an acute triangle.

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