Question

the equilateral triangle with side 12 has altitude
select one:
a. $6\sqrt{3}$
b. 24
c. $6\sqrt{2}$
d. 6

a square is inscribed in a circle. a tangent is drawn to the vertex. what size angle is formed?
select one:
a. 90
b. 180
c. 45
d. 22.5

Explanation:

First Question:

Step1: Split triangle into 2 right triangles

The altitude divides the equilateral triangle into two congruent right triangles, each with hypotenuse $12$, one leg $\frac{12}{2}=6$, and the other leg as the altitude $h$.

Step2: Apply Pythagorean theorem

Use $a^2 + b^2 = c^2$, where $c=12$, $a=6$, $b=h$.
$$h^2 + 6^2 = 12^2$$
$$h^2 = 144 - 36 = 108$$
$$h = \sqrt{108} = 6\sqrt{3}$$

Step1: Identify circle and square properties

A square inscribed in a circle has its diagonal as the circle's diameter. The tangent at a vertex is perpendicular to the radius at that point.

Step2: Calculate the target angle

The angle between the tangent and the square's side: the radius to the vertex bisects the square's 90° vertex angle (45°). The tangent is perpendicular to the radius (90°), so the angle between tangent and square side is $90^\circ - 45^\circ = 45^\circ$.

Answer:

a. $6\sqrt{3}$

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Second Question: