Find solutions to your questions with the help of IDNLearn.com's expert community. Discover comprehensive answers to your questions from our community of experienced professionals.

Type the correct answer in the box.

A printer creates a right triangular card where the hypotenuse, [tex]\( h \)[/tex], is three times as long as the shorter leg. What is the length of the longer leg in terms of [tex]\( m \)[/tex]? Write the answer in simplest form.


Sagot :

To find the length of the longer leg of a right triangle where the hypotenuse [tex]\( h \)[/tex] is three times as long as the shorter leg, labeled as [tex]\( m \)[/tex], we follow these steps:

1. Identify the sides of the triangle:
- The shorter leg is [tex]\( a = m \)[/tex].
- The hypotenuse is [tex]\( h = 3m \)[/tex].

2. Use the Pythagorean theorem, which states [tex]\( a^2 + b^2 = c^2 \)[/tex], to find the length of the longer leg [tex]\( b \)[/tex]:

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

Given:
[tex]\[ a = m, \quad c = 3m \][/tex]

Substitute these into the Pythagorean theorem:

[tex]\[ m^2 + b^2 = (3m)^2 \][/tex]

Simplify the right side:

[tex]\[ m^2 + b^2 = 9m^2 \][/tex]

Isolate [tex]\( b^2 \)[/tex] by subtracting [tex]\( m^2 \)[/tex] from both sides:

[tex]\[ b^2 = 9m^2 - m^2 \][/tex]

Simplify the equation:

[tex]\[ b^2 = 8m^2 \][/tex]

Take the square root of both sides to solve for [tex]\( b \)[/tex]:

[tex]\[ b = \sqrt{8m^2} \][/tex]

Simplify the square root:

[tex]\[ b = \sqrt{4 \cdot 2 \cdot m^2} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{m^2} = 2 \cdot \sqrt{2} \cdot m = 2\sqrt{2}m \][/tex]

Therefore, the length of the longer leg in terms of [tex]\( m \)[/tex] is:

[tex]\[ \boxed{2\sqrt{2}m} \][/tex]