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
  • medium-small < 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
    • SPECIAL CASE 1:
    • hypotenuse = largest side​
      • ​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 assignment was an open ended assignment given in four parts 

Each group needs to: 
  • find the missing hypotenuse
  • write a range of values of a side that is larger than the hypotenuse that would still produce a triangle
  • write a range of values of a side that is larger than the hypotenuse that would still produce a triangle
  • classify the types of triangles that NON-hypotenuse lengths of a missing side of a triangle
Concepts:
  • Right triangles
  • Oblique triangles 
  • Pythagorean Theorem
  • Triangle Inequality Theorem 
Resources Web page Links
Exploration 1
Row 1
  • 3/4
  • ​5/12
  • ​9/40
  • 7/25
Row 2
  • 9/11
  • 14/50
  • 27/123
  • 15/25
CLICK ON DESMOS LINK BELOW TO CHECK ANSWERS
Click on lower right-hand 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
  • 9/15
  • ​15/39
  • ​18/82
  • 30/50
Row 2
  • 6/8
  • ​6/12
  • ​18/20
  • 14/48
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
  • 15/20
  • ​50/100
  • ​18/24
  • 27/45
Row 2
  • 27/36
  • ​75/180
  • ​36/160
  • 21/75
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
  • 300/400
  • ​10/24
  • ​45/200
  • 175/625
Row 2
  • 8/15
  • ​24/51
  • ​45/90
  • 18/24
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
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