Right triangle ABC is divided into five identical right triangles as

Question

https://gmatclub.com/forum/download/file.php?id=61114 Right triangle ABC is divided into five identical right triangles as shown above. What is the ratio of AC to BC? A. \sqrt{5} : 2 B. 2\sqrt{5} : 5 C. \sqrt{5} : 1 D. \sqrt{3} : 1 E. 3 : 2\sqrt{3} 1.jpg

ShahadatHJ · Answer

Could someone please explain this one?

vipulgoel · Answer

here we go

let the diagonal of bigger tringle meets at AC at O

now it is given bigger right tringle is divided in 5 equal right tringles

5*1/2*OA * BO = 1/2 *OB *AC

AC = 5 OA

AC = OA + 2 identical Hight of smaller tringle , Hight of smaller tringle (OB) = 2 OA

now in ABO AB^2 = AO^2 + OB^2, AB = root 5

BC: AC = 2root 5 : 5 = root5 :2

Fdambro294 · Answer

(1st) the problem is based on visualization. The Longer Legs of each smaller, individual triangle have to be considered the same length. This is based on eyeballing which sides match up. Probably not a great idea on the actual test, but for this problem it is the only way to answer it.

(2nd) whenever a questions asks us for a ratio, we can choose a smart number for one side (given the constraints and the figure) that makes the calculation of the Ratio easier - then fill in the remaining lengths based on the constraints and figure out the relative ratio units

Let the hypotenuse of each smaller triangle = 2 ratio units

AB = one hypotenuse of a smaller triangle = 2 ratio units

BC = two hypotenuses of smaller triangles = 2 + 2 = 4 ratio units

Start from the left most smallest Right Triangle, and start matching up Angles as “X” and “90 - X” (across from the appropriate sides that look visually equal in length)

Again, the figure is based on visualization so place one angle label across from the longer Leg and one angle label across from the shorter Leg for each of the 5 triangles

At vertex B, you will have 1 of each of these angles:

Entire Angle at vertex B = (x) + (90 - x) = 90

The entire triangle ABC is a right triangle:

We made one Leg AB = 2 ratio units

And

The other Leg BC is thus = 4 ratio units

We can use the Pythagorean Theorem to find the Hypotenuse = AC

(AB)^2 + (BC)^2 = (AC)^2

(2)^2 + (4)^2 = (AC)^2

(AC)^2 = 20

AC = sqrt(20) = 2 * sqrt(5) ratio units

Finally:

AC / BC =   / 4 =

sqrt(5) : 2

Posted from my mobile device

IanStewart · Answer

There are a few ways to do this. You can notice, for example, that the big triangle must be similar to each identical small right triangle, because the big and small triangles all have one right angle, and the big triangle shares another angle with a small triangle (in two places in fact, at A and at C) so all three angles in the big triangle must be the same as the three angles in any small triangle. Then we can just answer the question using any of the small triangles, knowing the ratio of sides in two similar triangles will match up.

Since it's just a ratio question, as fdambro points out, we can make up a number, so we could call a short side of a small right triangle '1'. Then we can see that the vertical line, from B downwards, is length 2. But looking at the small triangle on the left, that vertical line is the other leg of our right triangle. So our short sides are in a 1 to 2 ratio, and if those are our actual lengths, the long side is, by Pythagoras, √5. If you use the similarity observation I mentioned above, the answer is now √5 to 2 immediately. If you didn't use that, you could notice that side AC is made up of one short leg and two of the other legs, so its length is 1 + 2 + 2 = 5. BC is made up of two hypotenuses, so its length is √5 + √5 = 2√5. So the answer is 5 to 2√5, or, dividing by √5, √5 to 2.

Right triangle ABC is divided into five identical right triangles as

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Right triangle ABC is divided into five identical right triangles as

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