Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In triangle ABC, D and E are points on AB and AC respectively such that DE||BC and DE divides triangle ABC into $2$ parts of equal areas. Find the ratio of AD and BD.

I am unable to start this question. Hope somebody could provide some hint.

share|improve this question
    
Hint: show the triangles ABC and ADE are similar. –  Gerry Myerson Sep 9 '13 at 10:12
    
but they are indeed similar. one parallel side, one same angle, two proportionate sides. But how would it lead me to solve the question? Hope you could guide further. –  Ramit Sep 9 '13 at 10:14
    
Second hint: How do the areas of similar triangle relate, if you know the scaling factor? –  Hagen von Eitzen Sep 9 '13 at 10:19
    
So, what does that say about a side of ADE, in relation to a corresponding side of ABC? –  Gerry Myerson Sep 9 '13 at 10:26
    
Let AG be the altitude of ABC, cutting DE at F. So, $\frac12*BC*AG=2*\frac12*DE*AF$. I am afraid it is not giving me anything. Certainly, $BC\ne2DE$, because had that been so, area of ADE would have been half that of trapezium DECB. –  Ramit Sep 9 '13 at 10:35
add comment

1 Answer 1

up vote 2 down vote accepted

First note that that the the triangles $ADE$ and $ABC$ are simular, because they have 3 parallel sides.

For two simular triangle the following statements hold:

$$\frac{a}{a_1} = \frac{b}{b_1} = \frac{c}{c_1} = \frac{h}{h_1} = k$$

$$\frac{P}{P_1} = k^2$$

Because the area of the smaller triangle is half of the bigger one we have:

$$\frac{2P}{P} = k^2$$ $$k^2 = 2$$ $$k = \sqrt{2}$$

Now from the first statement we have:

$$\overline{AB} = \overline{AD} \cdot k$$ $$\overline{AD} + \overline{BD} = \overline{AD} \cdot \sqrt{2}$$ $$\overline{BD} = (\sqrt{2} - 1)\overline{AD}$$ $$\frac{\overline{BD}}{\overline{AD}} = \sqrt{2} - 1$$ $$\frac{\overline{AD}}{\overline{BD}} = \frac{1}{\sqrt{2} - 1}$$

share|improve this answer
    
what are the smaller letters a b and c? what is P represent? is the h height? k looks like a constant! great please explain the relationships –  user2095034 Mar 6 at 21:58
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.