Difficult questions using basic concepts

It may seem like we will need trigonometry to handle this question, but that is not so. In fact, the question will look familiar at first, but will present unforeseen problems later on. While going through this exercise, we will learn a few tips and tricks which will be useful in our mainstream questions, hence, it will add value to our repertoire (especially in elimination techniques). Let’s go on to the question now: In triangle ABC, if angle ABC is 30 degrees, AC = 2*sqrt(2) and AB = BC = X, what is the value of X? (A) Sqrt(3) – 1 (B) Sqrt(3) + 2 (C) (Sqrt(3) – 1)/2 (D) (Sqrt(3) + 1)/2 (E) 2*(Sqrt(3) + 1) What we see here is an isosceles triangle with one angle as 30 degrees and other two angles as (180 – 30)/2 = 75 degrees each. The side opposite the 30 degrees angle is 2*sqrt(2). One simple observation is that X must be greater than 2*sqrt(2) because these sides are opposite the greater angles (75 degrees). 2*sqrt(2) is a bit less than 2*1.5 because Sqrt(2) = 1.414. So 2*sqrt(2) is a bit less than 3. Note that options (A), (C), and (D) are much smaller than 3, so these cannot be the value of X. We have already improved our chances of getting the correct answer by eliminating three options! Now we have to choose out of (B) and (E).                 Here is what is given: Angle ABC = 30 degrees, and AC = 2*sqrt(2). We need to find the value of X. Now, our 30 degree angle reminds us of a 30-60-90 triangle in which we know the ratio of the sides – given one side, we can find the other two. The problem is this: if we drop an altitude from angle B to AC, the angle 30 degrees will be split in half and we will actually get a 15-75-90 triangle, instead. We won’t have a 30-60-90 triangle anymore, so what do we do now? Let’s try to maintain the 30 degree angle as it is to get the 30-60-90 triangle, and drop an altitude from angle C to AB instead, calling it CE. Now we have a 30-60-90 triangle! Since BCE is a 30-60-90 triangle, its sides are in the ratio 1:sqrt(3):2. Side X corresponds to 2 on the ratio, so CE = x/2. Area of triangle ABC = (1/2)*BD*AC = (1/2)*CE*AB (1/2)*BD*2*sqrt(2) = (1/2)*(X/2)*X BD = X^2/4*Sqrt(2) Now DC = (1/2)AC = 2*sqrt(2)/2 = sqrt(2) Let’s use the pythagorean theorem on triangle BDC: BD^2 + DC^2 = BC^2 (X^2/4*Sqrt(2))^2 + (Sqrt(2))^2 = X^2 X^4/32 + 2 = X^2 X^4 – 32*X^2 + 64 = 0 X^4 – 16X^2 + 8^2 – 16X^2 = 0 (X^2 – 8)^2 – (4X)^2 = 0 (X^2 -8 + 4X) * (X^2 – 8 – 4X) = 0 Normally, this would require us to use the quadratic roots formula, but let’s not get that complicated. We can just plug in the the two shortlisted options and see if either of the factors is 0. If one of the factors becomes 0, the equation will be satisfied and we will have the root of the equation. Since both options have both terms positive, it means the co-efficient corresponding to B in Ax^2 + Bx + C = 0 must be negative. x = [-B +- Sqrt(B^2 – 4AC)]/2A -B will give us a positive term if B is negative, so we will get the answer by plugging into (X^2 – 4X – 8): Put X = Sqrt(3) + 2 in X^2 – 4X – 8 and you do not get 0. Put X = 2*(Sqrt(3) + 1) in X^2 – 4X – 8 and you do get 0. This means that X is 2*(Sqrt(3) + 1), so our answer must be (E). To recap: Tip 1: A greater side of a triangle is opposite a greater angle. Tip 2: We can get the relation between sides and altitudes of a triangle by using the area of the triangle formula. Tip 3: The quadratic formula can help identify the sign of the irrational roots.