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Seeing a lot of 'em and doing a lot of thinking. Also, not giving up. That may sound unpractical but it's pretty true - works for me. :-)
Fill in the missing parts of the following proof that if two triangles are similar, then the perimeters are in the same ratios as the corresponding sides. |dw:1360778980119:dw|Its geometry for tri angles
What do you know about the relative sizes of corresponding sides in similar triangles? And what do you know about the perimeter of a triangle? Here's a hint: think of the distributive property of multiplication: \[a(b+c+d) = ab + ac + ad\]
well it says r=a/b=b/e=c/f is def. of proportional (common ratio) Its Proofs and im just so llost :(
Don't you mean \[r = \frac ad = \frac be = \frac cf\]?
I'm sure you must :-) Take \[r=\frac ad\]What if you already knew the values of \(r\) and \(d\)? Could you find the value of \(a\)? If so, how?
sorry my computer is really slow. They dont give out #s just letters because i have 2 prove perimeter of (triangle)ABC is proportional to (triangel)DEF I have to give legit resons in steps like an investigatoon but i dont know wat to write :(
I understand that. If you had numbers, could give find the value of \(a\)? Or could you find the value with just variables, if you were willing to have an answer that contained variables?
Im sorry im really confused :(
Okay, you have an equation relating the lengths of sides a and d, do you not? \[r = \frac ad\] Can you rearrange that to give me the value of \(d\) if you know \(r\) and \(a\)?
Let's pretend that we know the values of all the sides of the small triangle \((a,b,c)\). We have that statement that \[r=\frac ad = \frac be = \frac cf\]We'll explore just the first part of that expression. \[r = \frac ad\]Multiply both sides by \(d\) and we get \[dr = a\]Divide by \(r\) and we have \[d = \frac ar\]Divide by \(a\) and we have \[\frac da = \frac 1r\] In other words, the ratio of side \(d\) to side \(a\) is \(1/r\). Now let's do that for the other sides, and we get \[e = \frac br\]\[f = \frac cr\] Do you see the pattern? Each of the sides of the sides of the big triangle can be gotten by multiplying the corresponding side of the small triangle by a constant, \(1/r\). Now the perimeter of the small triangle is just the sum of its sides, \((a+b+c)\). What is the perimeter of the big triangle? Again, it is the sum of its sides, \((d + e + f)\). What happens if you write the perimeter of the big triangle using those expressions we got for \(d, e, f\)? \[(\frac ar + \frac br + \frac cr)\]Looks pretty interesting! What if we factor out \(1/r\)? (This is just using the distributive property of multiplication in reverse) \[\frac1r(a+b+c)\]but \((a+b+c)\) is just the perimeter of the small triangle, and that factor of \(1/r\) in front is just the ratio of any side in the small triangle to the corresponding side in the big triangle! So, we've established that if the sides are proportional, the perimeter is proportional as well.