Math  /  Geometry

Question「层了 The sides of a triangle have lengths 21,29 , and 31 . What kind of triangle is acute right obtuse

Studdy Solution

STEP 1

What is this asking? We need to figure out if a triangle with sides of length 2121, 2929, and 3131 is acute, right, or obtuse. Watch out! Don't just guess based on how the triangle might look in your head!
We need to use the Pythagorean Theorem (or a close relative) to be absolutely sure.

STEP 2

1. Check for a Right Triangle
2. Classify the Triangle

STEP 3

First, we need to find the **longest side**, which will be our potential hypotenuse if this *is* a right triangle.
Looking at our sides, 2121, 2929, and 3131, the longest side is clearly 3131.

STEP 4

Let's see if the Pythagorean Theorem holds true.
Remember, the theorem states that in a right triangle, a2+b2=c2a^2 + b^2 = c^2, where cc is the **hypotenuse** (longest side) and aa and bb are the other two sides.
So, we'll check if 212+29221^2 + 29^2 equals 31231^2.

STEP 5

Let's calculate those squares! 212=2121=44121^2 = 21 \cdot 21 = 441, 292=2929=84129^2 = 29 \cdot 29 = 841, and 312=3131=96131^2 = 31 \cdot 31 = 961.

STEP 6

Now, let's add the squares of the shorter sides: 441+841=1282441 + 841 = 1282.
Is this equal to the square of the longest side?
Nope! 12821282 is *not* equal to 961961.

STEP 7

Since 212+292=128221^2 + 29^2 = 1282 is *greater* than 312=96131^2 = 961, this tells us something important!
When the sum of the squares of the two shorter sides is **greater** than the square of the longest side, the triangle is **acute**!

STEP 8

The triangle with sides 2121, 2929, and 3131 is an **acute** triangle!

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