Of course! Let's solve this step-by-step.
1. Understand the Problem:
We have a right triangle where:
- [tex]\( b = 3.3 \)[/tex] kilometers (one leg)
- [tex]\( c = 5 \)[/tex] kilometers (the hypotenuse)
2. Use the Pythagorean Theorem:
The Pythagorean theorem states that for a right triangle:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
3. Rearrange to Solve for [tex]\( a^2 \)[/tex]:
Substitute in the given values [tex]\( b = 3.3 \)[/tex] and [tex]\( c = 5 \)[/tex]:
[tex]\[
a^2 + 3.3^2 = 5^2
\][/tex]
[tex]\[
a^2 + 10.89 = 25
\][/tex]
4. Solve for [tex]\( a^2 \)[/tex]:
[tex]\[
a^2 = 25 - 10.89
\][/tex]
[tex]\[
a^2 = 14.11
\][/tex]
5. Find [tex]\( a \)[/tex]:
Take the square root of both sides to find [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt{14.11}
\][/tex]
[tex]\[
a \approx 3.75
\][/tex]
6. Calculate the Perimeter:
The perimeter [tex]\( P \)[/tex] of a triangle is the sum of all its sides:
[tex]\[
P = a + b + c
\][/tex]
Substitute [tex]\( a \approx 3.75 \)[/tex], [tex]\( b = 3.3 \)[/tex], and [tex]\( c = 5 \)[/tex]:
[tex]\[
P = 3.75 + 3.3 + 5
\][/tex]
[tex]\[
P = 12.05
\][/tex]
7. Round to the Nearest Tenth:
Therefore, rounding [tex]\( 12.05 \)[/tex] to the nearest tenth:
[tex]\[
P \approx 12.1
\][/tex]
Final Answer:
The perimeter of the right triangle is approximately [tex]\( 12.1 \)[/tex] kilometers.
