relationship between an angle and the length of its side

[bocu.alina]

Hello,

I came across this question in the Challenge Questions dated 6/4/07, the answer is D.

My question is: since sides DB and CD are in a ratio of 1:2, shouldn't the angles DAB and CAD be in a ratio of 1:2 as well? In this case since angle BAD is 15 dgrs shouldn't angle CAD be 30 dgrs?

Thank you in advance!

Question
In the figure to the right, point D divides side BC of triangle ABC into segments BD and DC of lengths 1 and 2 units respectively. Given that ADC = 60° and ABD = 45°, what is the measure of angle x? (Note: Figure is not drawn to scale.)

(A) 55°
(B) 60°
(C) 70°
(D) 75°
(E) 90°

[adiagr]

Hello,

I came across this question in the Challenge Questions dated 6/4/07, the answer is D.

My question is: since sides DB and CD are in a ratio of 1:2, shouldn't the angles DAB and CAD be in a ratio of 1:2 as well? In this case since angle BAD is 15 dgrs shouldn't angle CAD be 30 dgrs?

Thank you in advance!

Question
In the figure to the right, point D divides side BC of triangle ABC into segments BD and DC of lengths 1 and 2 units respectively. Given that ADC = 60° and ABD = 45°, what is the measure of angle x? (Note: Figure is not drawn to scale.)

(A) 55°
(B) 60°
(C) 70°
(D) 75°
(E) 90°

Hi,

two points.

1. All such ratios have to be applied in one triangle. Here the two sides you are considering are from 2 different triangles.

2. sides are in the ratios of Sine of respective angles i.e.

for e.g. in the given figure:

in triangle ABC:

(AC / Sin 45) = (AB/ Sin x)

[mittuetc]

Can anyone who knows the answer, please post the detailed explanation of the solution?

Thanks a lot.

[adiagr]

Can anyone who knows the answer, please post the detailed explanation of the solution?

Thanks a lot.

in triangle ABC

(BC/ Sin CAB) = (AC / Sin 45)

in triangle ADC

(DC/ Sin CAD) = (AC / Sin 60)

DC/BC= 2/3............see figure

Sin CAB / Sin CAD = (3/2)(Sin 45/ sin 60)

(Sin CAB / Sin CAD) = (Sin 60/ Sin 45)

angle CAB = 60 deg

angle X = 75 deg (triangle ABC, sum of angles to be 180 deg)


Ans is D.

[mittuetc]

Thanks a lot for the detailed explanation.

Could you please point me to the link or notes about the application of the formula itself? That is, I am new to this formula and would like to know the details about what is the formaula and when to apply this formula ...sides are in the ratios of Sine of respective angles.

Greatly appreciate your time and effort.

[mittuetc]

Can someone please help me by providing the details of what and when to apply this formula?

Thanks in advance

[RonPurewal]

My question is: since sides DB and CD are in a ratio of 1:2, shouldn't the angles DAB and CAD be in a ratio of 1:2 as well? In this case since angle BAD is 15 dgrs shouldn't angle CAD be 30 dgrs?

nope. that's not how angles work -- angles are not found in the same proportion as sides. in fact, that will almost never happen, unless you're dividing an isosceles triangle right down the middle (so that the proportion in each case is 1:1).

--

to the other posters, i've posted a copy of the official solution here:
http://www.postimage.org/image.php?v=gxLB7vr

--

by the way, this problem is way, way, WWWAAAAAAAAAYYYYY harder than anything you will ever see on the gmat. so, if you can't solve it, this is not any sort of negative against you.

--

adiagr --

i've got to hand it to you, that's a very clever solution.

however, for posters who don't already know trigonometry as well as adiagr does, it's not worth the time to study trigonometry rules -- you won't be able to use them unless the terms equate in such a fashion that you don't actually have to evaluate the trigonometric expressions, as happens in adiagr's solution. (note that adiagr doesn't have to find the value of either sin 45 or sin 60, since both of these expressions appear on both sides of the equation.)

the only angles for which you can really compute trig functions in your head are 0, 30, 45, 60, 90, and related angles -- but you don't need trig functions for these angles, since the requisite proportions are already covered by your memorized templates for the 30-60-90 and 45-45-90 triangles.