Question

unit 4 - introduction to polygons and triangles
you are the captain of the starship selecsoi from the planet yrtemoeg. while traveling in a
galaxy far, far away, you spotted a previously uncharted red dwarf star at a line of sight of 45°
from the ship (a). you needed to chart the new star, but you couldnt determine its distance
from the ship because the star was out of sensor range. after some quick thinking, you
remembered something you learned in geometry that helped you figure out a way to find the
distance. as the ship continued on its course (\\( \overline { a b } \\)), you kept watching the star until the line of
sight was exactly 90° from the ship (b). the distance you traveled from point a to point b was
125 million retemoliks. from this information, you were able to calculate the distance (\\( \overline { b c } \\)) to
the star and chart its position.

a. first, use the triangle sum theorem to calculate \\( m \angle c = \\)
b. based on that, what kind of triangle is \\( \delta a b c \\) ?
c. what does that tell you about the measure of \\( \overline { b c } \\) ?
d. therefore, the distance to the star from b to c was

Explanation:

Step1: Calculate \(m\angle C\)

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is \(180^{\circ}\). In \(\triangle ABC\), \(\angle A = 45^{\circ}\), \(\angle B=90^{\circ}\). Let \(m\angle C=x\). Then \(45^{\circ}+90^{\circ}+x = 180^{\circ}\). Solving for \(x\):
\(x=180^{\circ}-(45^{\circ}+90^{\circ})=45^{\circ}\)

Step2: Determine the type of \(\triangle ABC\)

Since \(\angle A = 45^{\circ}\) and \(\angle C = 45^{\circ}\), the triangle has two equal angles. A triangle with two equal angles is an isosceles triangle. Also, since \(\angle B = 90^{\circ}\), it is a right - angled triangle. So, \(\triangle ABC\) is a right - isosceles triangle.

Step3: Analyze the measure of \(\overline{BC}\)

In a right - isosceles triangle (\(\triangle ABC\) with \(\angle A=\angle C = 45^{\circ}\)), the legs are equal. That is, if \(\overline{AB}\) and \(\overline{BC}\) are the legs (in a right - triangle, the sides adjacent to the right - angle are legs), then \(BC = AB\)

Step4: Find the length of \(\overline{BC}\)

Given \(AB = 125\) million retemoliks. Since \(BC = AB\) (from the property of a right - isosceles triangle), \(BC=125\) million retemoliks

Answer:

a. \(45^{\circ}\)
b. Right - isosceles triangle
c. \(BC = AB\)
d. \(125\) million retemoliks