Hello!

I am having trouble understanding this question from my CAT test #1.

Is there a rule about dividing triangles and how they end up to be similar? i.e. I am having trouble seeing why and how the triangles are similar.

Also, in the first sentence of the solution, it says: we are given a right triangle that is cut into 4 smaller triangles, however I only see 3.

If you can shed some more light on the solution, that would be greatly appreciated.

Thank you!




In the diagram to the right, what is the length of AB?
(1) BE = 3
(2) DE = 4
(D) Each Statement Alone is Sufficient
We are given a right triangle that is cut into four smaller right triangles. Each smaller triangle was formed by drawing a perpendicular from the right angle of a larger triangle to that larger triangle's hypotenuse. When a right triangle is divided in this way, two similar triangles are created. And each one of these smaller similar triangles is also similar to the larger triangle from which it was formed.
Thus, for example, triangle ABD is similar to triangle BDC, and both of these are similar to triangle ABC. Moreover, triangle BDE is similar to triangle DEC, and each of these is similar to triangle BDC, from which they were formed. If BDE is similar to BDC and BDC is similar to ABD, then BDE must be similar to ABD as well.
Remember that similar triangles have the same interior angles and the ratio of their side lengths are the same. So the ratio of the side lengths of BDE must be the same as the ratio of the side lengths of ABD. We are given the hypotenuse of BDE, which is also a leg of triangle ABD. If we had even one more side of BDE, we would be able to find the side lengths of BDE and thus know the ratios, which we could use to determine the sides of ABD.
(1) SUFFICIENT: If BE = 3, then BDE is a 3-4-5 right triangle. BDE and ABD are similar triangles, as discussed above, so their side measurements have the same proportion. Knowing the three side measurements of BDE and one of the side measurements of ABD is enough to allow us to calculate AB.

To illustrate:
BD = 5 is the hypotenuse of BDE, while AB is the hypotenuse of ABD.
The longer leg of right triangle BDE is DE = 4, and the corresponding leg in ABD is BD = 5.

Since they are similar triangles, the ratio of the longer leg to the hypotenuse should be the same in both BDE and ABD.
For BDE, the ratio of the longer leg to the hypotenuse = 4/5.
For ABD, the ratio of the longer leg to the hypotenuse = 5/AB.
Thus, 4/5 = 5/AB, or AB = 25/4 = 6.25

(2) SUFFICIENT: If DE = 4, then BDE is a 3-4-5 right triangle. This statement provides identical information to that given in statement (1) and is sufficient for the reasons given above.

Any time you draw an altitude from a right angle, you create 2 smaller triangles each of which is similar to the original triangle. An altitude is specifically a line that creates a new right angle and, for this rule to work, the altitude has to be drawn from a right angle as well.

In the diagram in this problem, an altitude is draw first via the vertical red line BD which divides ABC into two new right triangles, ABD and BDC, both of which are similar to ABC.

Then, another altitude is drawn from D to E, which divides BDC into two new right triangles, BDE and DEC, both of which are similar to BDC. Since BDC is also one of the triangles in the previous step, that means all four triangles (the big one and the three little, interior ones) are similar.

If you want to know why this rule is true, compare the parts of the triangles you create. Let's do the first split (from ABC to ABD and BDC). First, all three triangles have a right angle, so that is one angle that is identical. Then, look at ABC and ABD. They also share the angle A - that has to be identical for both. Now, they have 2 identical angles, angle A and a right angle. Since all 3 angles in a triangle always add to 180, then the third angles in each must also be identical. If all three angles have the same measure, the two triangles are similar - that's the definition of a similar triangle. This is the proof for the AA or angle-angle theorem for proving similar triangles: if you can show that two angles are identical, then the two triangles are similar.

Hi,

In the above traingles I dont understand which traingle is similar to which traingle , I realize that they are similar but I could not understand how

"triangle ABD is similar to triangle BDC, and both of these are similar to triangle ABC. Moreover, triangle BDE is similar to triangle DEC, and each of these is similar to triangle BDC, from which they were formed. If BDE is similar to BDC and BDC is similar to ABD, then BDE must be similar to ABD as well"

To begin with how do I know that ABD is similar to BDC and not ADB is not similar to BDC

My confusion is after realizing that the right angle comes in the middle I dont know the order of the other two sides.. If they share a common side and a right angle I know the orde but given only right angles for two traingles and nothing abt the remaining sides I cannot fine the order..Thanks

as for the first question - regarding which triangles are similar to which other triangles - the answer, nicely enough, is that ALL of the triangles in the above diagram are similar.
in general, when you take a right triangle and draw an altitude (perpendicular segment from the opposite corner) down to the hypotenuse, you generate a set of triangles in which each triangle is similar to each other triangle.

to justify this fact, try to relate each triangle back to triangle ABC (the biggest triangle in the diagram).
looking at triangle ABD, we see that it shares (literally) angle A with triangle ABC, and that both ABD and ABC have right angles. that's two angles that match, which is enough to prove that the triangles are similar. (the third angles must match as well, because they're equal to 180 minus the first two angles.) so, in order, the similarity is ABC ~ ADB.
same argument works for all the other triangles: just notice an angle that's shared with the larger triangle, and then notice that there's a right angle in both triangles.

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