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805+ Level|   Geometry|      
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raj44
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 Updated on: 12 Dec 2014, 07:43
Originally posted by raj44 on 12 Dec 2014, 07:10.
Last edited by Bunuel on 12 Dec 2014, 07:43, edited 1 time in total.
Renamed the topic and edited the question.
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E
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95% (hard)

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24% (01:56) correct 76% (02:18) wrong based on 174 sessions

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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 40 degrees. What is the value of angle DAB?

(1) AD is angular bisector of angle A.
(2) AB x CD = AC x BD
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E
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viktorija
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 12 Dec 2014, 15:03
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Can anybody explain why the answer is E?
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raj44
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 13 Dec 2014, 00:33
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viktorija
Can anybody explain why the answer is E?
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The answer is E because the question stem doesn't specify which 2 angles are equal, angle A or angle C could both be equal to 40. Also, the 2 statements mean the same- AD as angle bisector itself means ab x bd = ac x dc
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 17 Dec 2014, 06:55
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viktorija
Can anybody explain why the answer is E?

The answer is E because the question stem doesn't specify which 2 angles are equal, angle A or angle C could both be equal to 40. Also, the 2 statements mean the same- AD as angle bisector itself means ab x bd = ac x dc
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Unable to understand how ab x bd = ac x dc means AD is angle bisector?

ab ac
--- = ----
bd dc

means ABD and ACD are simliar triangles , not same triangles.
Sides are in proportion, does it mean angles are same?

Kindly clarify.
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 17 Dec 2014, 08:59
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raj44
viktorija
Can anybody explain why the answer is E?

The answer is E because the question stem doesn't specify which 2 angles are equal, angle A or angle C could both be equal to 40. Also, the 2 statements mean the same- AD as angle bisector itself means ab x bd = ac x dc

Unable to understand how ab x bd = ac x dc means AD is angle bisector?

ab ac
--- = ----
bd dc

means ABD and ACD are simliar triangles , not same triangles.
Sides are in proportion, does it mean angles are same?

Kindly clarify.
Show more

It's the angle bisector theorem. Check Wiki here: https://en.wikipedia.org/wiki/Angle_bisector_theorem
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 25 May 2015, 02:04
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 40 degrees. What is the value of angle DAB?

(1) AD is angular bisector of angle A.
(2) AB x CD = AC x BD
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Bunuel,
I have one query here.
Does the angle bisector extends 90 degree angle at base?
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Bunuel
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 25 May 2015, 02:17
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ssriva2
raj44
In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 40 degrees. What is the value of angle DAB?

(1) AD is angular bisector of angle A.
(2) AB x CD = AC x BD



Bunuel,
I have one query here.
Does the angle bisector extends 90 degree angle at base?
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Sorry, I don't understand what you mean...
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 25 May 2015, 03:06
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I hope its clear now.i have attached figure for same
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File comment: Does angle bisector extends 90 degree on the hypotenuse
IMG_20150525_153013_AO_HDR.jpg
IMG_20150525_153013_AO_HDR.jpg [ 1.33 MiB | Viewed 7201 times ]

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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 25 May 2015, 03:12
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ssriva2
I hope its clear now.i have attached figure for same
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Only for for an isosceles triangle (or for an equilateral), the angle bisector to base coincides with the height.
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 26 May 2015, 08:11
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ssriva2
I hope its clear now.i have attached figure for same
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Hi ssriva2,

You can also understand this property in the following manner. Refer the diagram below:



In the diagram, ABC is an isosceles triangle with sides AB = AC. As angles opposite the equal sides are equal we have ∠B = ∠C = \(x\). Also AD is the angle bisector of ∠A, therefore ∠BAD = ∠CAD = \(\frac{y}{2}\).

We can also see from the above figure that ∠BDA = ∠CDA = \(z\) (as ∠BDA = ∠CDA = \(180 - x - \frac{y}{2}\))

We also know that in triangle ABC, \(2x + y = 180\). Similarly in triangle ABD we can write \(z + x + \frac{y}{2} = 180\) i.e. \(2z + 2x + y = 360\).

Since \(2x + y = 180\), we have \(2z + 180 = 360\) i.e. \(z = 90\).

However please note that the only the angle bisector of the non-equal angle of an isosceles triangle will be perpendicular to the opposite base i.e. the non-equal side. Also AD bisects base BC i.e. BD = DC. These properties will not apply to the equal angles and the equal sides.

In case of an equilateral triangle, we know that \(x = 60\) and \(\frac{y}{2} = 30\), hence \(z = 90\).
In an equilateral triangle since all angles and sides are equal, these properties would apply to any of the angles and sides.

Hope it's clear :)

Regards
Harsh
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In an Isosceles Triangle ABC, point D lies on segment BC. Angle B is 4 05 Nov 2017, 03:07
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ssriva2
I hope its clear now.i have attached figure for same

Hi ssriva2,

You can also understand this property in the following manner. Refer the diagram below:



In the diagram, ABC is an isosceles triangle with sides AB = AC. As angles opposite the equal sides are equal we have ∠B = ∠C = \(x\). Also AD is the angle bisector of ∠A, therefore ∠BAD = ∠CAD = \(\frac{y}{2}\).

We can also see from the above figure that ∠BDA = ∠CDA = \(z\) (as ∠BDA = ∠CDA = \(180 - x - \frac{y}{2}\))

We also know that in triangle ABC, \(2x + y = 180\). Similarly in triangle ABD we can write \(z + x + \frac{y}{2} = 180\) i.e. \(2z + 2x + y = 360\).

Since \(2x + y = 180\), we have \(2z + 180 = 360\) i.e. \(z = 90\).

However please note that the only the angle bisector of the non-equal angle of an isosceles triangle will be perpendicular to the opposite base i.e. the non-equal side. Also AD bisects base BC i.e. BD = DC. These properties will not apply to the equal angles and the equal sides.

In case of an equilateral triangle, we know that \(x = 60\) and \(\frac{y}{2} = 30\), hence \(z = 90\).
In an equilateral triangle since all angles and sides are equal, these properties would apply to any of the angles and sides.

Hope it's clear :)

Regards
Harsh
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So based on this itself why is it E? why not D?
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