need help with a geometry problem (altitude and geometric mean)

[Lucy the Valiant]

Help!

My 2 students and I all missed problem #4. I've attached a photo because I don't know how to draw it here, and we have spent most of the morning watching Khan and Mr. Hammy's explanations, but still cannot figure out how to solve it since we do not have the whole hypotenuse. 

Given: RW = 3 and ST = 20/3.   Find RT.

 

We understand that the altitude SW is the geometric mean of the hypotenuse and its nearest leg, such that WT / ST = ST / RT. Also, RW / RS = RS / ST. 

 

But then, no matter which way we manipulate it, we think that we need to know WT! 

 

 

[Lucy the Valiant]

Nobody? Girls and I think it's a typo, and that they actually intended to give us WT (or RT). 

 

This was a test, and they got all the other problems correct, so . . . maybe? (fingers crossed? that it's a typo?) 

[Jann in TX]

One side of the original triangle is the geometric mean between the whole hypotenuse and the 'part' of the hypotenuse it touches.

So in your example ST is the Geometric mean between RT and WT

let WT = x

20/3 over (x+3) = x over 20/3

9x^2 + 27x -400 = 0

x = 16/3   This is WT's  measure add 9/3 to get RT's measure 25/3

[Lucy the Valiant]

41 minutes ago, Jann in TX said:

One side of the original triangle is the geometric mean between the whole hypotenuse and the 'part' of the hypotenuse it touches.

So in your example ST is the Geometric mean between RT and WT

let WT = x

20/3 over (x+3) = x over 20/3

9x^2 + 27x -400 = 0

x = 16/3   This is WT's  measure add 9/3 to get RT's measure 25/3

 

Oh, Jann, THANK YOU! 

We kept getting 9x^2 + 27x -400 = 0, but because NONE of the problems in the book used a quadratic, we (mistakenly) concluded that we were going about it incorrectly and didn't even bother to factor it out. 

Argh. This makes sense. THANK YOU, and we will press on! 

(I was really hoping you were going to tell me it's a typo, LOL!)