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# One more

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In the diagram, triangle ABE, triangle BCE and triangle CDE are right-angled, with angle AEB=angle BEC = angle CED = 60 degrees, and AE=24.

Find the length of CE.

Thank you so much!

AnonymousConfusedGuy  Feb 24, 2018

#1
+2248
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The problem is easier than I first thought.

By the given information, we know that there are three right-angled triangles in the diagram. We know that $$m\angle AEB=m\angle BEC=m\angle CED= 60^{\circ}$$. We can use this information to determine that every right triangle is also a 30-60-90 triangle. We also know that $$AE=24$$.

A 30-60-90 triangle is a special kind of right triangle where the ratio of the side lengths are $$1:\sqrt{3}:2$$$$\overline{AE}$$ is the longest side length because it is the hypotenuse of the largest right triangle in the diagram. $$\overline{BE}$$ is the shortest side length of $$\triangle ABE$$ because this side is opposite the smallest angle. We mentioned earlier what the ratio of the side lengths are, so we can determine the length of $$\overline{BE}$$ without doing anything too computationally demanding.

 $$\frac{BE}{AE}=\frac{1}{2}$$ We already know the ratio of the side lengths of a 30-60-90 triangle, so we can apply this relationship and create a proportion. We already know what AE equals, so let's fill that in. $$\frac{BE}{24}=\frac{1}{2}$$ In order to solve a proportion, simply cross multiply. $$2BE=24$$ Divide by 2 on both sides to determine the unknown length of the side. $$BE=12$$

Of course, the ultimate goal is to figure out the length of $$\overline{CE}$$. If you look at $$\triangle BCE$$, carefully, you will notice that we are in an identical situation to when we solved for $$BE$$. Notice that $$\overline{BE}$$ is the hypotenuse of this triangle, and $$\overline{CE}$$ is the shortest side length since it is opposite the 30º angle. We can use the same $$1:\sqrt{3}:2$$ relationship of the side lengths to find the missing length.

 $$\frac{CE}{BE}=\frac{1}{2}$$ Just like before, we know what the value of BE is, so let's plug it in! $$\frac{CE}{12}=\frac{1}{2}$$ Just like before, cross multiplying is the way to go! $$2CE=12$$ Divide by 2 on both sides to solve this problem. $$CE=6$$
TheXSquaredFactor  Feb 24, 2018
#1
+2248
+2

The problem is easier than I first thought.

By the given information, we know that there are three right-angled triangles in the diagram. We know that $$m\angle AEB=m\angle BEC=m\angle CED= 60^{\circ}$$. We can use this information to determine that every right triangle is also a 30-60-90 triangle. We also know that $$AE=24$$.

A 30-60-90 triangle is a special kind of right triangle where the ratio of the side lengths are $$1:\sqrt{3}:2$$$$\overline{AE}$$ is the longest side length because it is the hypotenuse of the largest right triangle in the diagram. $$\overline{BE}$$ is the shortest side length of $$\triangle ABE$$ because this side is opposite the smallest angle. We mentioned earlier what the ratio of the side lengths are, so we can determine the length of $$\overline{BE}$$ without doing anything too computationally demanding.

 $$\frac{BE}{AE}=\frac{1}{2}$$ We already know the ratio of the side lengths of a 30-60-90 triangle, so we can apply this relationship and create a proportion. We already know what AE equals, so let's fill that in. $$\frac{BE}{24}=\frac{1}{2}$$ In order to solve a proportion, simply cross multiply. $$2BE=24$$ Divide by 2 on both sides to determine the unknown length of the side. $$BE=12$$

Of course, the ultimate goal is to figure out the length of $$\overline{CE}$$. If you look at $$\triangle BCE$$, carefully, you will notice that we are in an identical situation to when we solved for $$BE$$. Notice that $$\overline{BE}$$ is the hypotenuse of this triangle, and $$\overline{CE}$$ is the shortest side length since it is opposite the 30º angle. We can use the same $$1:\sqrt{3}:2$$ relationship of the side lengths to find the missing length.

 $$\frac{CE}{BE}=\frac{1}{2}$$ Just like before, we know what the value of BE is, so let's plug it in! $$\frac{CE}{12}=\frac{1}{2}$$ Just like before, cross multiplying is the way to go! $$2CE=12$$ Divide by 2 on both sides to solve this problem. $$CE=6$$
TheXSquaredFactor  Feb 24, 2018
#2
+1442
+1

Thanks so much!

AnonymousConfusedGuy  Feb 26, 2018