Question

Describe three ways to determine the measure of segment YZ.

Answer 1

Using 3 different methods, we can find that YZ = 25m

For a given triangle rectangle with a known angle θ and a hypotenuse H, we can write the trigonometric relations

sin(θ) = (opposite cathetus)/H

cos(θ) = (adjacent cathetus)/H

tan(θ) = (opposite cathetus)/(adjacent cathetus)

Now, in the given image we can see that:

θ = 30°

H = 50m

XY = adjacent cathetus

YZ = opposite cathetus.

We want to find YZ, so with the known things, we can use the first relation:

$sin(30\°) = YZ/50m$

$sin(30\°)*50m = YZ = 25m$

Now let's try another way.

We know that the sum of the internal angles of a triangle is always equal to 180°

In a triangle rectangle, we always have an angle equal to 90°.

Then the missing angle for our triangle can be computed from:

Z + 90° + 30° = 180°

Z = 180° - 90° - 30° = 60°

From this angle, the side YZ is the adjacent cathetus, then we can use the second trigonometric relation:

$cos(60\°) = YZ/50m\\cos(60\°)*50m = YZ = 25m$

Third way.

Whit the same angle we could find the side XY, the opposite cathetus, with:

$sin(60\°) = XY/50m \\sin(60\°)*50m = XY = 43.3m$

Now that we know one of the catheti, we can use the last trigonometric relation

$tan(60\°) = XY/YZ = 43.3m/YZ \\YZ = 43.3m/tan(60\°) = 25m$

If you want to learn more about triangle rectangles, you can read:

brainly.com/question/24330420