## In triangle $$ABC$$ above, what is the length of side $$BC?$$

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### In triangle $$ABC$$ above, what is the length of side $$BC?$$

by Vincen » Sat Nov 27, 2021 4:48 am
In triangle $$ABC$$ above, what is the length of side $$BC?$$

(1) Line segment $$AD$$ has length $$6.$$
(2) $$x = 36$$

Source: Official Guide

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### Re: In triangle $$ABC$$ above, what is the length of side $$BC?$$

by [email protected] » Sat Nov 27, 2021 7:30 am
Vincen wrote:
Sat Nov 27, 2021 4:48 am
Untitled.png

In triangle $$ABC$$ above, what is the length of side $$BC?$$

(1) Line segment $$AD$$ has length $$6.$$
(2) $$x = 36$$

Source: Official Guide
Target question: What is the length of side BC?

Statement 1: Line segment AD has length 6.
BEFORE we deal with statement 1, let's see what information we can add to the diagram.

For example, since ∆BDC has 2 equal angles (of 2x°), we know that side BD = side BC:

Next, since angles on a line add to 180°, and since ∠BDC = 2x°, we know that ∠ADB = (180 - 2x)°

Since angles in a triangle add to 180°, we know that ∠ABD = x°
ASIDE: Notice that x° + x° + (180 - 2x)° = 180°

Now that we know ∆BAD has two equal angles (x° and x°), we know that side AD = side BD

This means AD = BD = BC

Statement 1 tells us that AD = 6, which means BC = 6
The answer to the target question is side BC has length 6
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = 36
Notice that our diagram doesn't any lengths.
We can SHRINK or ENLARGE the diagram and the angles remain the same.
However the length of side BC changes.

Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT