For triangle ABC, angle ABC = 90 degrees, and side AC has a length of

IMPORTANT: For geometry DS questions, we are typically checking to see whether the statements "lock" a particular angle or length into having just one value. This concept is discussed in much greater detail in our video at the bottom of this post.

This technique can save a lot of time.

Target question: What is the length of line segment BD?

Given: For triangle ABC, angle ABC = 90 degrees, and side AC has a length of 15.

So, we have a shape that looks something like this . . .

. . . where the legs of the triangle can vary AND the location of point D can vary.

Statement 1: Triangle ABC is isosceles.
Since there is ONLY ONE isosceles right triangle with hypotenuse 15, this statement LOCKS triangle ABC into having one and only one shape.


However, statement 1 does NOT lock in the location of point D.
Since this statement does not lock in the location of point D, the length of BD is NOT LOCKED IN.
Consider these two examples.


Notice the different lengths of line segment BD
Since statement 1 does not lock in the length of line segment BD, it is NOT SUFFICIENT

Statement 2: Line segment BD is perpendicular to side AC.
This statement locks in the location of point D (in relation to the triangle's hypotenuse), but it does NOT lock in the shape of the triangle.
Consider these two examples:


Notice the different lengths of line segment BD
Since statement 2 does not lock in the length of line segment BD, it is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 locks in the shape of triangle ABC.
Statement 2 then locks in the location of point D as follows:

Since there's only one diagram that can be drawn with the given information, there can be ONLY ONE length of line segment BD
Are we required to find this length? No. We need only recognize that there can be only one length.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer =

RELATED VIDEO

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ans C.. a right angle triangle given with length of hyp...
1) it tells us the other two sides are equal and gives the length of other two sides.. however D can be anywhere on line AC ..insufficient
2)it just tells us that BD is altitude.. no use unless sides are known..insufficient

combined sufficient as allsides known and alt bd can be found out
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Statement 1: We know that side AB and side BC are of the same length.
However, the segment BD can be of various lengths based on where the point D is located along side AC.
Insufficient.

Statement 2: segment BD forms a 90 deg angle with side AC. However, we still do not know how side AC is broken down or what the length is of any other side.
Insufficient.

Combined, we know right triangle ABC is isosceles with BD as its altitude and bisector. We can now apply pythagorean theorem to find the length of segment BD.

Answer: C

I start with (2): since we do not know
the length of AD and CD we can not use the similar triangle theorem, (sid/alt=alt/side)
not suff.
(1)triangle is a isosceles. but we do not know where is the point D on AC.

(1/2): since BD is perpendicular, we know that the isosceles ABC with °ABC=90 has BCA=45 and CAB=45. with AC = 15 we got AD=7,5 and CD=7,5. We could either use x:x:x*root(2) to determine AB / CB and afterwars BD (with pythagoras).
or we know that the perpendicular will build 2 triangles with 45-45-90 but as the ground we will get x, and as the legs we will get 0,5*15. so its suff

Answer C
1) If ABC is isosceles AB=BC but doesnt help us calculate BD.
2) If BD is perpendicular to AC we still dont know BD.

Combining, we know AB = BC from one so each of them can be calculated and the triangle is defined. So BD is calculable.
Or since area of triangle is constant AB*BC=AC*BD so BD can be calculated.

Answer =C.

Press kudos if I am right.

VERITAS PREP OFFICIAL SOLUTION:

C. For statement 1, it's helpful to just draw the triangle and several variations of line BD - you should see that point D can be extremely close to point A, right down the middle, or extremely close to point C, all with different lengths. This demonstrates that statement 1 is not sufficient.

For statement 2, recognize that triangle ABC and be a right angle with an extremely long side AB and an extremely short side BC, or (as alluded to in statement 1) it could be isosceles in which both sides are the same length. And in either case, the length of line BD will differ dramatically.

If both statements are taken together, however, you can find the length of BD. Since ABC is an isosceles right triangle with a hypotenuse of 15, sides AB and BC will each have a length equal to 15 divided by the square root of 2. And since line BD will bisect triangle ABC into two identical right triangles, each with either side AB or BC as the hypotenuse, you should note that side BD will have to equal the other "short" leg of each triangle (either AD or DC), and each of those is half of the long side AC, which is 15. So the length of BD is half of 15, which is 7.5
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Hi Bunuel,

I have understood your explanation. But I need one clarification.
What If I say, statement 2 alone is sufficient. Because we know hypotenuse is 15 and ABC is a right angled triangle and right angled at B. with Pythagoras tripplets, other two sides are 12 and 9. We know if we know 3 sides we can find out BD length.

with formula BD = (AB * BC) / AC.

With this I feel something I am missing. either I am assuming more than the required or carrying extra information. Can you tell me what is my mistake ?

Why do you assume that the sides should be Pythagorean Triples? Why do you assume that the sides should be integers at all?
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I almost concluded D as the answer by confusing similarity for congruency and then realised that strict congruency can only be reached if apart from being isosceles and having BD as the common side BD is also a perpendicular bisector of the AC. But this condition was not found in statement 1.
Then i checked statement 2 and realised that statement 2 gives us exactly what was missing .. the condition for strict congruency and the both statement A and B are needed

Attached here is my wrong answer

THE CORRECT ANSWER IS C


Attached here is my WRONG answer



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Responding to a pm:

This is where you made a mistake: there is no SSA rule. The rule is SAS (two sides and the INCLUDED angle). If you say AB = BC and BD = BD, the angles which have to be same are DBA and DBC.
A special SSA rule works only in case of a right triangle - It is called the RHS rule. There has to be a right angle in both triangles, the hypotenuses have to be equal and any one set of sides should be equal.

Statement 1 gives you neither SAS nor RHS. So the two triangles may not be congruent.
The other congruency rules are SSS, ASA and AAS.
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Many thanks Karishma

I realised I am arriving at that wrong answer if I am choosing D.
Just got confused about congruency.
You are a tremendous help..
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Responding to a pm:


Triangles ABC and ADB are similar using AA rule (two equal angles) - no problem there.
The point is - how do you get the length of the line segment BD?
All you know is that AC is 15. You don't know the ratio of the corresponding sides of the two triangles. You don't know AB/BC/AD. How will you find BD?
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Hi Bunuel
Just curious which is the above highlighted formulae. Can u plz share its applicability.Thanks

Hello

Consider a triangle ABC, which is 90 degrees at B. Now area of this triangle = 1/2 * base * height = 1/2 * AB * BC (AB, BC as base/height)
Now lets draw a perpendicular BD to side AC. Taking AC as base, BD would be the height. So Area can also be = 1/2 * BD * AC

Equating the two, AB*BC = BD*AC or BD = (AB * BC) / AC

You can find the area of a righ triangle ABC (right angled at B) in several ways.

1. Area = leg1*leg2/2 = AB*BC/2
2. Area = height*base/2 = BD*AC/2 (where BD is altitude from B to the hypotenuse and hypotenuse AC is the base).

Equate: AB*BC/2 = BD*AC/2;

AB*BC = BD*AC;

BD = AB*BC/AC

Hope it's clear.
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For triangle ABC, angle ABC = 90 degrees, and side AC has a length of 15. If point D lies on side AC, and a line is drawn from point B to point D, what is the length of line segment BD?


(1) Triangle ABC is isosceles.
Insufficient b/c we don't know where specifically D is located on AC.

(2) Line segment BD is perpendicular to side AC.
Insufficient b/c we don't know what type of triangle we are looking at.

Combined: Sufficient.

We have a 45-45-90 triangle and D is the midpoint of AC.
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