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**+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

#1**+2 **

Best Answer

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