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The lengths of two sides of a right triangle are 12 inches and 15 inches. What is the difference between the two possible lengths of the third side of the triangle? Round your answer to the nearest tenth.


___ inches

Respuesta :

calculista

we know that

In a right triangle

Applying the Pythagorean Theorem

[tex]c^{2} =a^{2} +b^{2}[/tex]

where

a and b are the legs of the triangle

c is the hypotenuse of the triangle

Step N [tex]1[/tex]

Assume that the third side is a leg

In this case we have

[tex]a=12\ in\\c=15\ in\\ b=?[/tex]

[tex]c^{2} =a^{2} +b^{2}[/tex]

Solve for b

[tex]b^{2}=c^{2} -a^{2}[/tex]

substitute the values

[tex]b^{2}=15^{2}-12^{2}[/tex]

[tex]b^{2}=81[/tex]

[tex]b=9\ in[/tex]

Step N [tex]2[/tex]

Assume that the third side is the hypotenuse

In this case we have

[tex]a=12\ in\\b=15\ in\\ c=?[/tex]

[tex]c^{2} =a^{2} +b^{2}[/tex]

substitute the values

[tex]c^{2}=12^{2}+15^{2}[/tex]

[tex]c^{2}=369[/tex]

[tex]c=19.2\ in[/tex]

Step N [tex]3[/tex]

Find the difference of the third sides

[tex]19.2\ in-9\ in=10.2\ in[/tex]

therefore

the answer is

[tex]10.2\ in[/tex]

The difference between the two possible lengths of the third side of a right triangle with two sides that are 12 inches and 15 inches is 10.2 inches.

Further Explanation:

Right triangle

  • A right triangle is a triangle with one of its angles being 90 degrees or right angle.
  • The triangle has two shorter sides making the right angle and the hypotenuse which is the longest side.

Pythagoras Rule

  • According to Pythagoras rule, in a right angled triangle if the squares of the shorter sides are added then they are equivalent to the square of the hypotenuse.
  • That is; [tex]a^{2} +b^{2} =c^{2}[/tex], where a and b are the shorter sides while c is the hypotenuse.

In this case;

  • We are given two sides whose lengths are 12 inches and 15 inches
  • Therefore; there are two possible lengths of the third sides where one is the shorter may be a hypotenuse.

First possibility

  • The third side is one of the shorter side.

Therefore;

a = 12 inches

b= ?

c = 15 inches

Solve for b

[tex]b^{2} = c^{2} - a^{2}[/tex]

Substituting the values

[tex]b^{2} = 15^{2}-12^{2}[/tex]

[tex]b^{2} = 81[/tex]

[tex]b = \sqrt{81}[/tex]

[tex]b = 9 Inches[/tex]

Second possibility

  • The third side is the hypotenuse

Therefore;

[tex]c^{2} = a^{2} + b^{2} \\[/tex]

Substituting the values

[tex]c^{2} = 15^{2} + 12^{2}[/tex]

[tex]c^{2} = 369[/tex]

[tex]c=\sqrt{369}[/tex]

[tex]c=19.2 Inches[/tex]

The difference between the two possible lengths

= 19.2 in - 9 in

= 10.2 inches

Keywords: Right triangle, Pythagoras rule

Learn more about:

  • Pythagoras theorem: https://brainly.com/question/6241673
  • Right triangle: https://brainly.com/question/12453859

Level; High school

Subject: Mathematics

Topic: Pythagoras theorem

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