Geometry Stratgy Guide 4th edition: Chpt 3, Pg 66 Problem 11

[Pnwankpa]

In this problem we are asked to find the value of "X" based on the figure drawn. We are presented with a clearly defined triangle with a height of 3, side 7, and side 12. The triangle attached this larger triangle has the height of "X" and an unknown base.
I attempted to solve, by using the Pythagorean theorem, however, I solved incorrectly and the explanation given on the answer page only confused me more.

Please help.
Detailed step-by-step explanation needed.

Thank you.

[tommywallach]

Hey Pn,

You can't really use pythagoras here. To do so you'd have to do the following: Set the unknown dotted length in the bottom right corner as y.

Now solve for the WHOLE right triangle, with height x and (7+y):

x^2 + (7+y)^2 = 144

49 + 14y + y^2 + x^2 = 144

NOW, use pythagoras to solve within the triangle:

3^2 + z^2 = 7^2
9 + z^2 = 49
z^2 = 40
z = rt. 40 = 2rt.10

That means the base of the other right triangle is (12 - 2rt. 10), so we can use pythagoras again:

3 ^2 + (12 - 2rt.10)^2 = w^2

You could then TECHNICALLY solve for w. Once you have w, you can write the equation for the final right triangle (made up of two dotted lines) as x ^ 2 + y ^ 2 = w ^ 2.

From there, you can replace the x^2 + y^2 in the FIRST equation we came up with (At the top of this post), and solve for y, then x.

Phew! Don't do it!

-t

[Pnwankpa]

Hello TommyWallace,

Thank you for showing another alternative to solving this problem. It seems to be an even longer process. It to me a minute to understand where you were headed, but I caught on.
Is it possible to explain this problem using the solution provided in the answer section (page 71) of the book?
I think what is keeping me from understanding the breakdown provided in the guide is that I am not seeing how 12 and 7 came to be used as bases, or the premise as to why it was solved the way it is in the book.

Thank you for replying.

Hey Pn,

You can't really use pythagoras here. To do so you'd have to do the following: Set the unknown dotted length in the bottom right corner as y.

Now solve for the WHOLE right triangle, with height x and (7+y):

x^2 + (7+y)^2 = 144

49 + 14y + y^2 + x^2 = 144

NOW, use pythagoras to solve within the triangle:

3^2 + z^2 = 7^2
9 + z^2 = 49
z^2 = 40
z = rt. 40 = 2rt.10

That means the base of the other right triangle is (12 - 2rt. 10), so we can use pythagoras again:

3 ^2 + (12 - 2rt.10)^2 = w^2

You could then TECHNICALLY solve for w. Once you have w, you can write the equation for the final right triangle (made up of two dotted lines) as x ^ 2 + y ^ 2 = w ^ 2.

From there, you can replace the x^2 + y^2 in the FIRST equation we came up with (At the top of this post), and solve for y, then x.

Phew! Don't do it!

-t

[tommywallach]

Heyo,

I can't improve on that explanation, if that's what you're asking. If you'd like to point out some specific place in the explanation where they lost you, let me know.

-t