Math  /  Geometry

QuestionQuadratic, Rational, and Radical Equations Pythagorean Theorem \square 1/3 Español ig right triangle, find the side length xx. Round your answer to the nearest hundredth

Studdy Solution

STEP 1

What is this asking? We need to find the length of the longest side of a right triangle, given the lengths of the other two sides. Watch out! Remember that the Pythagorean Theorem only works for right triangles!
Also, make sure we're solving for the right side – the *hypotenuse* – which is opposite the right angle.

STEP 2

1. Set up the Pythagorean Theorem
2. Solve for xx

STEP 3

Alright, let's **start** with the **Pythagorean Theorem**: a2+b2=c2a^2 + b^2 = c^2.
Remember, aa and bb are the lengths of the two shorter sides (legs) of the right triangle, and cc is the length of the longest side (hypotenuse).
In our triangle, the legs have lengths 55 and 77, and the hypotenuse has length xx.

STEP 4

Let's **plug in** our values.
We can set a=5a = 5 and b=7b = 7, and since xx is the hypotenuse, we have c=xc = x.
So, our equation becomes: 52+72=x25^2 + 7^2 = x^2

STEP 5

Let's **simplify** those squares: 25+49=x225 + 49 = x^2

STEP 6

Now, **add** those numbers together: 74=x274 = x^2

STEP 7

To **isolate** xx, we need to take the **square root** of both sides of the equation.
Remember, when we take the square root of a number, we get both a positive and a negative result.
However, since we're dealing with a length, we only care about the positive value. 74=x2\sqrt{74} = \sqrt{x^2} x=74x = \sqrt{74}

STEP 8

Now, let's **calculate** that square root and **round** to the nearest hundredth: x8.60x \approx 8.60

STEP 9

The length of the side xx is approximately **8.60**.

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