Sure, let's solve the problem step-by-step.
### Part (a): Find the Hypotenuse
To find the hypotenuse in a right triangle when the lengths of the two legs are given, we use the Pythagorean theorem. The theorem is stated as follows:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
where [tex]\(c\)[/tex] is the hypotenuse, and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the legs of the triangle.
Given:
[tex]\[
a = 5.00
\][/tex]
[tex]\[
b = 8.00
\][/tex]
Substitute these values into the Pythagorean theorem:
[tex]\[
c^2 = 5.00^2 + 8.00^2
\][/tex]
Calculate the squares:
[tex]\[
c^2 = 25.00 + 64.00
\][/tex]
[tex]\[
c^2 = 89.00
\][/tex]
Now, take the square root of both sides to find [tex]\(c\)[/tex]:
[tex]\[
c = \sqrt{89.00} \approx 9.43
\][/tex]
So the hypotenuse is approximately [tex]\(9.43\)[/tex] (rounded to two decimal places).
### Part (b): Find the Two Acute Angles
To find the acute angles in a right triangle, we can use trigonometric functions. Specifically, we use the arctangent (inverse tangent) function, which relates the ratio of the legs to the angles.
Let [tex]\(\theta_1\)[/tex] and [tex]\(\theta_2\)[/tex] be the two acute angles.
1. Calculate [tex]\(\theta_1\)[/tex]:
[tex]\[
\theta_1 = \arctan\left(\frac{a}{b}\right)
\][/tex]
[tex]\[
\theta_1 = \arctan\left(\frac{5.00}{8.00}\right)
\][/tex]
Using a calculator to find the arctan and then converting it to degrees:
[tex]\[
\theta_1 \approx 32.0^\circ
\][/tex]
2. Calculate [tex]\(\theta_2\)[/tex]:
[tex]\[
\theta_2 = \arctan\left(\frac{b}{a}\right)
\][/tex]
[tex]\[
\theta_2 = \arctan\left(\frac{8.00}{5.00}\right)
\][/tex]
Using a calculator to find the arctan and then converting it to degrees:
[tex]\[
\theta_2 \approx 58.0^\circ
\][/tex]
Thus, the two acute angles are approximately [tex]\(32.0^\circ\)[/tex] and [tex]\(58.0^\circ\)[/tex] (rounded to one decimal place).
### Final Answer:
a) The hypotenuse is [tex]\(9.43\)[/tex].
b) The two acute angles are [tex]\(32.0^\circ\)[/tex] and [tex]\(58.0^\circ\)[/tex].
