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slwarrior64
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How can the perpendicular diagonals of a triangle help prove HCD?
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In summary, the triangles $BDE$ and $FDC$ are similar, as are $DCE$ and $DBF$. The little triangle $DEG$ is also similar to $BDE$ and $FDC$. Using this similarity, it can be deduced that both triangles $DHC$ and $DHF$ are isosceles. The result follows easily from the fact that $GDE$ and $DFH$ are equal angles and the intersection of lines $EF$ and $GH$. The triangle $CDF$ having a right angle at $D$ also helps to show that $HCD$ has equal angles at $C$ and $D$, making it also isosceles.
- #2
slwarrior64
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I have so far that triangles BDE and FDC are similar, as are DCE and DBF
- #3
Opalg
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The little triangle $DEG$ is also similar to $BDE$ and $FDC$. Use that to deduce that both of the triangles $DHC$ and $DHF$ are isosceles. The result should then follow quite easily.
- #4
slwarrior64
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One quick question, is DEG similar to BDE because we know both have a right angle, and then they share a side and they share an angle? Is that enough?Opalg said:
- #5
skeeter
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two equal angles are enough to show similarity
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slwarrior64
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True, I was thinking congruence. Thanks everyone!skeeter said:
- #7
slwarrior64
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Thank you!Opalg said:
- #8
slwarrior64
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I'm sorry, I'm definitely missing something similar, but how do I know they are isosceles?Opalg said:
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Opalg
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The angles $GDE$ and $DFH$ are equal (from the similar triangles $GDE$ and $DFC$).slwarrior64 said:
The angles $GDE$ and $FDH$ are equal (from the intersection of the lines $EF$ and $GH$).
Therefore the angles $DFH$ and $FDH$ are equal and so the triangle $FDH$ is isosceles. So the sides $HF$ and $HD$ must be equal.
Using the fact that the triangle $CDF$ has a right angle at $D$, you can then show that the triangle $HCD$ has equal angles at $C$ and $D$ and is therefore also isosceles.
- #10
slwarrior64
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Thanks, I was able to get triangle FDH, but I was stuck on proving HCD because I didn't know how that line would cut the right angleOpalg said:
- #11
slwarrior64
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I got it! Thanks again!slwarrior64 said:
FAQ: How can the perpendicular diagonals of a triangle help prove HCD?
What are perpendicular diagonals?
Perpendicular diagonals are two lines that intersect at a right angle and bisect each other, forming four right angles at the point of intersection.
How are perpendicular diagonals used in geometry?
Perpendicular diagonals are used to identify and construct shapes such as squares and rectangles. They also help to determine the properties of these shapes, such as equal side lengths and right angles.
What is the relationship between perpendicular diagonals and symmetry?
In symmetrical shapes, the perpendicular diagonals are equal in length and bisect each other at the center, creating two congruent triangles on either side. This is known as the axis of symmetry.
Can perpendicular diagonals exist in three-dimensional shapes?
Yes, perpendicular diagonals can exist in three-dimensional shapes, such as cubes and rectangular prisms. In these shapes, the perpendicular diagonals form a cross-section that divides the shape into four equal parts.
What is the practical application of understanding perpendicular diagonals?
Understanding perpendicular diagonals is important in fields such as architecture and engineering, where precise measurements and angles are necessary for constructing buildings and structures. It is also used in navigation and mapping to determine distances and angles between points.