Math  /  Geometry

QuestionIn a right triangle, aa and bb are the lengths of the legs and cc is the length of the hypotenuse. If a=3.4a=3.4 kilometers and b=1.8b=1.8 kilometers, what is cc ? If necessary, round to the nearest tenth. c=c= \square kilometers

Studdy Solution

STEP 1

What is this asking? We've got a right triangle, know the lengths of its legs, and need to find the length of the hypotenuse, rounding to the nearest tenth if needed. Watch out! Don't mix up the legs and the hypotenuse!
The hypotenuse is the longest side, opposite the right angle.

STEP 2

1. Recall the Pythagorean Theorem
2. Substitute and Solve
3. Round if Necessary

STEP 3

Alright, so we're dealing with a right triangle and need to relate the lengths of its sides.
The Pythagorean Theorem is *perfect* for this!
It says that in a right triangle, the sum of the squares of the lengths of the legs (aa and bb) is equal to the square of the length of the hypotenuse (cc).

STEP 4

So, our trusty theorem looks like this: a2+b2=c2a^2 + b^2 = c^2

STEP 5

We know that a=3.4a = \textbf{3.4} km and b=1.8b = \textbf{1.8} km.
Let's **plug these values** into the Pythagorean Theorem: (3.4)2+(1.8)2=c2(\textbf{3.4})^2 + (\textbf{1.8})^2 = c^2

STEP 6

Now, let's **square those values**: 11.56+3.24=c211.56 + 3.24 = c^2

STEP 7

**Add** those results together: 14.8=c214.8 = c^2

STEP 8

To find cc, we need to take the **square root** of both sides: 14.8=c2\sqrt{14.8} = \sqrt{c^2} c=14.8c = \sqrt{14.8}

STEP 9

Calculating the square root gives us: c3.847c \approx 3.847

STEP 10

The problem asks us to round to the nearest tenth.
Looking at our result, c3.847c \approx \textbf{3.847}, the digit in the hundredths place is 4, which is less than 5.
So, we round *down*.

STEP 11

Therefore, our **final answer** rounded to the nearest tenth is approximately 3.8\textbf{3.8} km.

STEP 12

c3.8c \approx \textbf{3.8} kilometers

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