Final answer:
The student's question involves drawing a geometric figure and applying principles like congruent triangles, trigonometric identities, law of sines, and law of cosines to solve for lengths and angles.
Step-by-step explanation:
The question involves creating a geometric figure based on the given conditions, which include two perpendicular line segments within a triangle, forming a rectangle.
Understanding the properties of rectangles, triangles, and the use of trigonometric and geometric principles are key to solving this problem.
To solve the problem, we must first draw triangle ABC with the given sides AB and AC.
Next, we construct MH perpendicular to AB at H, and MI perpendicular to AC at I, to form rectangle AHMI within triangle ABC.
Since AHMI is a rectangle, opposite sides are equal, so AH = MI and AM = HI.
By applying geometric principles, such as congruent triangles and trigonometric identities, we can find the lengths and angles specified in the question.
Additionally, properties like the law of sines and the law of cosines may be used to calculate unknown values within the figure.