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If angle BAD is a right angle, what is the length of side

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Re: If angle BAD is a right angle, what is the length of side [#permalink]

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New post 18 May 2016, 05:53
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komalsudhir wrote:
Bunuel Thanks for your reply. Median from Angle BAD will divide the hypotenuse BD into two halves. And, from statement 1 we know that AC is perpendicular to BD so AC satisfies both conditions of being a perpendicular and a bisector. Therefore from these two conditions we know that Triangle BAD is isosceles triangle. Therefore, side AB is also 5 and it makes side BD as 5√2.


How do you know for (1) that AC is the median?
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Re: If angle BAD is a right angle, what is the length of side [#permalink]

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New post 08 Nov 2016, 14:42
here is my two cents
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Re: If angle BAD is a right angle, what is the length of side [#permalink]

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New post 29 May 2017, 17:11
chiragatara wrote:
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If angle BAD is a right angle, what is the length of side BD?

(1) AC is perpendicular to BD

(2) BC = CD

[Reveal] Spoiler:
Official answer explanation:

Using statements 1 and 2, we know that AC is the perpendicular bisector of BD. This means that triangle BAD is an isosceles triangle so side AB must have a length of 5 (the same length as side AD). We also know that angle BAD is a right angle, so side BD is the hypotenuse of right isosceles triangle BAD. If each leg of the triangle is 5, the hypotenuse (using the Pythagorean theorem) must be 5 underroot 2.

Can someone kindly explainthe underlined portion?


The goal here is to find the length of side BD. Since it's DS, the problem is not drawn to scale.

Statement 1) AC is perpendicular to BD.

We don't know if BC = BD, so this statement is insufficient. There are many possible triangles here that would give a different length of BD.

Statement 2) BC = CD

We don't know if AC is perpendicular to BD, so knowing that the two sides are equal doesn't help us.

Statements 1+2) We know that AC is the perpendicular bisector of BD, so that means that the line AC forms two isosceles triangles (ACD and ACB). Since the sides of the height is equal to their bases and two each other, then that means the two triangles are congruent. Since they are congruent, BAD is an isosceles triangle and its base, the hypotenuse of the triangle in this case, is 5 \sqrt{2}
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Re: If angle BAD is a right angle, what is the length of side [#permalink]

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New post 08 Dec 2017, 00:14
Kekki wrote:
Isn't it that in an right angle triangle a perpenidcular line to hupothenuses (BC) halves the 90 degree triangle into 45° and 45° degrees?


Hi Bunuel,

Small question here

Since the AC divides the triangle into three similar triangles. and we have side AD as 5. Cant we calculate ACD as a pyth triplet, hence 3-4-5 and then
AC/BC=CD/AC. This would give us BC??
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Re: If angle BAD is a right angle, what is the length of side [#permalink]

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New post 08 Dec 2017, 02:19
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RedIndian wrote:
Kekki wrote:
Isn't it that in an right angle triangle a perpenidcular line to hupothenuses (BC) halves the 90 degree triangle into 45° and 45° degrees?


Hi Bunuel,

Small question here

Since the AC divides the triangle into three similar triangles. and we have side AD as 5. Cant we calculate ACD as a pyth triplet, hence 3-4-5 and then
AC/BC=CD/AC. This would give us BC??


Are you talking about the first statement? If yes, then are you saying that ACD is 3-4-5 right triangle? HOW do you know that? The fact that the length of a hypotenuse is 5 does NOT mean that it must necessarily be 3-4-5 type.
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New to the Math Forum?
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Resources:
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If angle BAD is a right angle, what is the length of side   [#permalink] 08 Dec 2017, 02:19
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