Certainly! Let's solve for the height using the Pythagorean theorem. We're given that the hypotenuse [tex]\(c\)[/tex] is 50 and the base [tex]\(b\)[/tex] is 48. We want to find the height [tex]\(a\)[/tex].
Step 1: Write the Pythagorean theorem in the form that relates the height, base, and hypotenuse:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Step 2: Substitute the given values [tex]\(c = 50\)[/tex] and [tex]\(b = 48\)[/tex] into the equation:
[tex]\[
a^2 + 48^2 = 50^2
\][/tex]
Step 3: Rearrange the equation to solve for [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 50^2 - 48^2
\][/tex]
Step 4: Calculate the values of [tex]\(50^2\)[/tex] and [tex]\(48^2\)[/tex]:
[tex]\[
50^2 = 2500
\][/tex]
[tex]\[
48^2 = 2304
\][/tex]
Step 5: Subtract [tex]\(48^2\)[/tex] from [tex]\(50^2\)[/tex]:
[tex]\[
a^2 = 2500 - 2304 = 196
\][/tex]
Step 6: Take the square root of both sides to solve for [tex]\(a\)[/tex]:
[tex]\[
a = \sqrt{196} = 14.0
\][/tex]
Therefore, the height is:
[tex]\[
\boxed{14.0}
\][/tex]
Additionally, we found that the value of the height squared is:
[tex]\[
\boxed{196}
\][/tex]
So, the height of the triangle is [tex]\(14.0\)[/tex] units, and the height squared is [tex]\(196\)[/tex].
