Question

For each given set of numbers, determine if a triangle with those side
lengths can be made or not. If a triangle can be made, determine if the
triangle is a right triangle. Justify all answers.
b.
8, 12,4
a.
8, 15, 17

Answer 1

A triangle with three set of sides that are Pythagorean Triple will make a right triangle, while a triangle with three sides that satisfy the Triangle Inequality Theorem will make a triangle. therefore:

• Sides 8, 12,4 will make any triangle
• Sides 8, 15, 17 will make a triangle and it is a right triangle.

Recall:

• The Pythagorean Triple is three sets of positive numbers that conforms to the criteria given as: $a^2 + b^2 = c^2$ (i.e., the sum of the squares of the two shorter sides of a triangle, a and b, must be equal to the square of the longest side, c).

• The sides of a right triangle are Pythagorean Triple.

• According to the Triangle Inequality Theorem, a any triangle can be made if the sum of any two sides is greater than the third side.

Therefore, let's check and see if 8, 12,4 or 8, 15, 17 will make a triangle, and also if the triangle is a right triangle.

Check if 8, 15, 17 will make a triangle by applying the Triangle Inequality Theorem:

8 + 15 > 17

23 > 17 (true)

17 + 15 > 8

32 > 8 (true)

17 + 8 > 15

25 > 15 (true)

• Therefore, a triangle with sides 8, 15, 17 can be made.

Determine if the triangle whose sides are 8, 15, 17 is a right triangle:

Where,

a = 8

b = 15

c = 17,

Substitute the values into  $a^2 + b^2 = c^2$ .

Thus:

$8^2 + 15^2 = 17^2\\\\64 + 225 = 289\\\\289 = 289$ (true)

• Therefore, 8, 15, 17 will make a right triangle.

Check if 8, 12,4 will make a triangle by applying the Triangle Inequality Theorem:

8 + 12 > 4

20 > 4 (true)

8 + 4 > 12

12 > 12 (False)

12 + 4 > 12

16 > 12 (true)

• Therefore, a triangle with sides 8, 12,4 cannot be made.

Determine if the triangle whose sides are 8, 15, 17 is a right triangle:

Where,

a = 4

b = 8

c = 12

Substitute the values into  $a^2 + b^2 = c^2$ .

Thus:

$4^2 + 8^2 = 12^2\\\\16 + 64 = 144\\\\80 = 144$ (False)

• Therefore, 8, 12,4 will not make a right triangle.

In summary, a

triangle with three set of sides that are Pythagorean Triple will make a right triangle, while a triangle with three sides that satisfy the Triangle Inequality Theorem will make a triangle. therefore:

• Sides 8, 12,4 will make any triangle
• Sides 8, 15, 17 will make a triangle and it is a right triangle.

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https://brainly.com/question/18754145