What if you had two sides of a right triangle?
If you are only told that you have a pair of sides of a right triangle and nothing else, then you have TWO possible right triangles
 leg/leg/unknown hypotenuse
 find the sum of the squares of the given sides
 take square root of the result
 simplify radicals when asked to and capable
 approximate when necessary
 unknown leg/leg/hypotenuse
 the hypotenuse MUST be the larger of the two given numbers!
 find the difference of the squares of the given sides
 take square root of the result
 simplify radicals when asked to and capable
 approximate when necessary
What if I have three sides of a triangle?
small/medium/large side was given
 mediumsmall < large < medium + small OR NOT A triangle
 This is a case/result of the Triangle Inequality Theorem
 if small and medium lead to hypotenuse, then four things could be true
 hypotenuse > largest side
 this means your triangle is an acute scalene triangle
 hypotenuse > largest side and largest side is smaller than small + medium sides
 this means that your triangle is an obtuse scalene triangle
 this means that your triangle is an obtuse scalene triangle
 SPECIAL CASE 1:
 hypotenuse = largest side
 this means your triangle is an acute scalene triangle
 this means your triangle is an acute scalene triangle
 SPECIAL CASE 2:
 hypotenuse > largest side and largest side is larger than small + medium sides
 this means that NO TRIANGLE EXISTS with these three side length
 this means that NO TRIANGLE EXISTS with these three side length
 hypotenuse > largest side
This assignment was an open ended assignment given in four parts
Each group needs to:

Concepts:

Resources Web page Links

Exploration 1
Row 1

Row 2

CLICK ON DESMOS LINK BELOW TO CHECK ANSWERS
Click on lower righthand corner to check your answers
Row 3
 6/8/9
 acute triangle because the largest side is smaller than its related right triangle's hypotenuse
 scalene triangle because three different side lengths
 best classification acute scalene
 6/8/10
 THIS IS THE RIGHT triangle
 Satisfies the Pythagorean Theorem's side relationships
 Best classification right scalene triangle
 6/8/11
 obtuse triangle because it is a triangle whose largest side exceeds the hypotenuse of the related right triangle
 why is it a triangle at all?
 the sum of 6 and 8 is 14 and the difference of 8 and 6 is 2
 2 < largest side < 14
 best classification obtuse scalene
 6/8/15
 NOT a triangle
 WHY?
 the sum of 6 and 8 is 14 and the difference of 8 and 6 is 2
 largest side > 14
Row 1 Support 3/4/missing hypotenuse
Row 1 Support missing leg/3/4
Exploration 2
Row 1

Row 2

Row 3
 9/12/13
 acute triangle because the largest side is smaller than its related right triangle's hypotenuse
 scalene triangle because three different side lengths
 best classification acute scalene
 9/12/15
 THIS IS THE RIGHT triangle
 Satisfies the Pythagorean Theorem's side relationships
 Best classification right scalene triangle
 9/12/18
 obtuse triangle because it is a triangle whose largest side exceeds the hypotenuse of the related right triangle
 why is it a triangle at all?
 the sum of 9 and 12 is 21 and the difference of 12 and 9 is 3
 3< largest side < 21
 best classification obtuse scalene
 9/12/21
 NOT a triangle
 WHY?
 the sum of 9 and 12 is 21 and the difference of 9 and 12 is 3
 largest side = 21
Exploration 3
Row 1

Row 2

Row 3
 12/16/19
 acute triangle because the largest side is smaller than its related right triangle's hypotenuse
 scalene triangle because three different side lengths
 best classification acute scalene
 12/16/20
 THIS IS THE RIGHT triangle
 Satisfies the Pythagorean Theorem's side relationships
 Best classification right scalene triangle
 12/16/22
 obtuse triangle because it is a triangle whose largest side exceeds the hypotenuse of the related right triangle
 why is it a triangle at all?
 the sum of 12 and 16 is 28 and the difference of 12 and 16 is 4
 4< largest side < 28
 best classification obtuse scalene
 12/16/29
 NOT a triangle
 WHY?
 the sum of 12 and 16 is 28 and the difference of 12 and 16 is 4
 largest side = 29
Exploration 4
Row 1

Row 2

Row 3
 4/12/13
 obtuse triangle because it is a triangle whose largest side exceeds the hypotenuse of the related right triangle
 why is it a triangle at all?
 the sum of 4 and 12 is 16 and the difference of 4 and 12 is 8
 8< largest side < 16
 best classification obtuse scalene
 10/24/25
 acute triangle because the largest side is smaller than its related right triangle's hypotenuse
 scalene triangle because three different side lengths
 best classification acute scalene
 15/36/39
 THIS IS THE RIGHT triangle
 Satisfies the Pythagorean Theorem's side relationships
 Best classification right scalene triangle
 20/48/52
 THIS IS THE RIGHT triangle
 Satisfies the Pythagorean Theorem's side relationships
 Best classification right scalene triangle
As a mathematics teacher, one person would have a hard time producing the amount of resources that the internet provides, so I supplement my resources with the assist of MANY other hard working and excellent mathematicians from the internet. I will try my best to give them credit and thank them anonymously in total here. I hope that others would visit and support their sites. I try and avoid those that are attempting to sell their services because I am not trying to induce a profit or monetary support personally, but I respect their right to do so, just that I am not trying to enter into ADVERTISEMENT or REFERRAL business.