Monday, June 13, 2016

Daily Regents: Right Triangle Altitude Theorem (January 2016)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams next week. At least, that is the plan.

I had a question about this problem on the thread where I posted all the multiple-choice answers: http://mrburkemath.blogspot.com/2016/02/january-2016-new-york-geometry-common.html

Common Core Geometry, January 2016, Question 22

30. 2 In the diagram below, CD is the altitude drawn to the hypotenuse AB of right triangle ABC.
Which lengths would not produce an altitude that measures 6√2?

The Right Triangle Altitude Theorem tells us that (AD)(DB) = (CD)2
The square of the altitude is (6√2)(6√2) = (36)(2) = 72.
So which choice has lengths of AD and DB that have a product of something other than 72?

Careful! Two of the options give you AB instead of DB. Subtract AB - AD to get DB.

Look at the four choices:

  1. - 2 * 36 = 72. Not the answer
  2. - 3 * (24 - 3) = 3 * 21 = 63, not 72. This is the correct answer.
  3. - 6 * 12 = 72. Not the answer
  4. - 8 * (17 - 8) = 8 * 9 = 72. Not the answer.

The correct answer is choice (2).


Any questions?

If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.


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2 comments:

  1. AnonymousFriday, June 09, 2023 7:26:00 PM

    From my perspective question 22 has no correct answer because the question does not refer to altitude CD; it just says produce "an altitude." There are three altitudes in any triangle. In this right triangle, its two legs (AC and BC) are altitudes, as well as CD. Using the values from choice (2), we could produce an altitude that measures 6*sqrt(2): AD = 3 and AB = 24 --> CD = 3*sqrt(7). Using the Pythagorean Theorem on triangle ADC, we find that AC = 6*sqrt(2). Thus, the lengths given in choice (2) do produce an altitude (i.e., altitude AC) that measures 6*sqrt(2).

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  2. (x, why?)Thursday, June 15, 2023 5:48:00 PM

    Feel free to write New York State and have them update the question in their archives. You never know -- they've done it before.

    One could argues that since they referenced altitude CD in the first place of the question that this is the altitude to which they are referring. Moreover, when one applies the Right Triangle Altitude Theorem, altitude CD is the one generally being referenced.

    That said, I don't think any students read so much into this problem that they believed that the legs of the right triangle couldn't be 6 sqrt(2) either.

    Thank you for stopping by the blog.

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