DO NOT ANSWER W/O EXPLANATION PLEASE! NEED WRITTEN SOLUTION. In triangle ABC, an altitude is drawn from vertex C to the line containin

Question

DO NOT ANSWER W/O EXPLANATION PLEASE! NEED WRITTEN SOLUTION.
In triangle ABC, an altitude is drawn from vertex C to the line containing AB. The length of this altitude is h and h=AB. Which of the following is true?
I. Triangle ABC could be a right triangle.
II. Angle C cannot be a right angle.
III. Angle C could be less than 45 degrees.

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Alexandra 4 days 2021-10-10T21:10:38+00:00 1 Answer 0

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    2021-10-10T21:12:00+00:00

    We have altitude h to side AB and AB=h, i.e. the altitude is congruent to the side it goes to.

    That’s all kinds of triangles.  One way to see them is using two horizontal parallel lines h apart, the bottom one with a base AB=h somewhere on it.  Then any C on the top line makes a triangle ABC with altitude h=AB.

    Let’s go through the choices.

    I. ABC could be a right triangle.  That’s TRUE.

    We could have the isoscleles right triangle, C directly above B, so AC is the leg and an altitude, AB=AC and B is the right angle.

    II. Angle C cannot be a right angle.  That’s TRUE.

    The biggest angle C can be is when it’s over the midpoint of AB, so if AB=2, h=2, and

    AC=\sqrt{2^2+1^2}=\sqrt{5}

    so

    C_{\textrm{max}} = 2 \arctan(1/2) \approx 53.13^\circ

    III. Angle C could be less than 45 degrees.   That’s TRUE.

    As long as C stays on our top parallel, we can make it as acute as we like by going farther away from AB.

    All true.  Hmmm.

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