Math  /  Geometry

QuestionWhat combinations DIDNT WORK to form a triangle? * 2,3,42,3,4 1,2,31,2,3 4,5,84,5,8 5,6,75,6,7 1,3,51,3,5 3,5,63,5,6 2,3,62,3,6 4,5,74,5,7 2,4,62,4,6 1,2,41,2,4

Studdy Solution

STEP 1

What is this asking? Which sets of three numbers *can't* make a triangle? Watch out! There's a sneaky rule about triangle side lengths, so don't get tripped up!

STEP 2

1. The Triangle Inequality Theorem
2. Check the Combinations

STEP 3

Alright, future geometers!
The **Triangle Inequality Theorem** says that the sum of any two sides of a triangle *must* be greater than the third side.
Think of it like this: if you're trying to make a triangle with sides that are too short, they won't be able to reach each other!
It'd be like trying to make a sandwich with two tiny slices of bread and a giant piece of ham – it just won't close!

STEP 4

So, mathematically, if we have a triangle with sides aa, bb, and cc, then ALL of these must be true:
a+b>ca + b > ca+c>ba + c > bb+c>ab + c > a

STEP 5

Let's test our number sets!
We'll check if they satisfy the **Triangle Inequality Theorem**.

STEP 6

{2,3,4}\{2, 3, 4\}: 2+3>42 + 3 > 4, 2+4>32 + 4 > 3, 3+4>23 + 4 > 2.
This one works!

STEP 7

{1,2,3}\{1, 2, 3\}: 1+2>31 + 2 > 3 is *not* true!
So, {1,2,3}\{1, 2, 3\} *cannot* make a triangle.

STEP 8

{4,5,8}\{4, 5, 8\}: 4+5>84 + 5 > 8 is *not* true!
Another one bites the dust!

STEP 9

{5,6,7}\{5, 6, 7\}: 5+6>75 + 6 > 7, 5+7>65 + 7 > 6, 6+7>56 + 7 > 5.
Triangle time!

STEP 10

{1,3,5}\{1, 3, 5\}: 1+3>51 + 3 > 5 is *not* true!
No triangle here.

STEP 11

{3,5,6}\{3, 5, 6\}: 3+5>63 + 5 > 6, 3+6>53 + 6 > 5, 5+6>35 + 6 > 3.
Triangle party!

STEP 12

{2,3,6}\{2, 3, 6\}: 2+3>62 + 3 > 6 is *not* true!
No triangle.

STEP 13

{4,5,7}\{4, 5, 7\}: 4+5>74 + 5 > 7, 4+7>54 + 7 > 5, 5+7>45 + 7 > 4.
Triangle!

STEP 14

{2,4,6}\{2, 4, 6\}: 2+4>62 + 4 > 6 is *not* true!
No triangle for you!

STEP 15

{1,2,4}\{1, 2, 4\}: 1+2>41 + 2 > 4 is *not* true!
No triangle again!

STEP 16

The sets that *can't* make a triangle are {1,2,3}\{1, 2, 3\}, {4,5,8}\{4, 5, 8\}, {1,3,5}\{1, 3, 5\}, {2,3,6}\{2, 3, 6\}, {2,4,6}\{2, 4, 6\}, and {1,2,4}\{1, 2, 4\}.

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