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Triangle ABC has a right angle at corner C. It has a height from C to a point D on side |AB|. If |CD|=5 and |AD|=7 then what is the length of the hypotenuse? (|AB|=?)

Correct Answer: 74/7

I have tried solving the question above by the help of the law of cosines and sines, pythagoras theorm and uniform triangles but without much success. I always tend to get to many unknown variables. Thanks in advance.

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  • $\begingroup$ right $\triangle$-s ACD and ACB are similar, since they share angle a. Therefore, $|AD|/|CD| = |AC|/|BC|. $\endgroup$ – user2661923 May 2 '18 at 23:17
  • $\begingroup$ @IGotAQuestion Please recall that if the OP is solved you can evaluate to accept an answer among the given, more details here meta.stackexchange.com/questions/5234/… $\endgroup$ – user May 31 '18 at 19:49
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Draw a picture. You just need similar triangles. $ABC,ACD,$ and $CBD$ are similar. $\frac {BD}{DC}=\frac {CD}{AD}=\frac 57$ so $BD=\frac {25}7$ and $AB=AD+BD=7+\frac {25}7=\frac {74}7$

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Remember that the height, $CD$, will be perpendicular to side $AB$. You can use Pythagorean Theorem to find length $AC$. Hint: Can you find the angle $\angle CBA$? This will help you.

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  • $\begingroup$ How can you know that the height CD will be perpendicular to AB? It seems to me to be an anjustified assumption. $\endgroup$ – IGotAQuestion May 2 '18 at 23:20
  • $\begingroup$ The height (or altitude) of a triangle referenced to side $b$ must be perpendicular to $b$. You could also look at the other answers. There are probably many ways of solving it. $\endgroup$ – D.B. May 2 '18 at 23:23
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HINT

  • make a sketch of the triangle
  • by Pytagoras find $AC$ from $AC^2=CD^2+AD^2$
  • then use similarity to find that $\frac{AB}{AC}=\frac{AC}{AD}$

enter image description here

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  • $\begingroup$ How can you know that the height CD will be perpendicular to AB? It seems to me to be an anjustified assumption. $\endgroup$ – IGotAQuestion May 2 '18 at 23:22
  • $\begingroup$ it is true by definition of height, the height from a vertex is always perpendicular to the opposite side en.wikipedia.org/wiki/Altitude_(triangle) $\endgroup$ – user May 2 '18 at 23:23
  • $\begingroup$ So if it says that a height is drawn from an angle, it is always perpendicular to the side it is drawn? Thanks! $\endgroup$ – IGotAQuestion May 2 '18 at 23:24
  • $\begingroup$ @IGotAQuestion Yes exactly, take a look to the link for the definition. $\endgroup$ – user May 2 '18 at 23:25

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