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3 years ago

In a 30-60-90 triangle, if the the side opposite 30 degrees has a length

Mathematics

1 answer:

son4ous [18]3 years ago

8 0

Might Want to draw this and Its very long but Hope it helped :)

Answer:

The sides are all of the same length - let's say a. The angles are all the same too, and since the angles must add up to 180∘, we conclude that the three angles in the equilateral triangle are equal to 180∘/3=60∘.

Now we do something sneaky. We draw a line all the way down from the top vertex of the triangle to the midpoint of the bottom line.

This new line cuts our equilateral triangle in half. What are the angles in one half?

The angle at the bottom is 90∘.

One of the angles is the same as one of the angles in the original equilateral triangle, so it is 60∘.

So the third angle must be 180∘−90∘−60∘=30∘.

Now the hypotenuse of this new triangle is a, the side length of the equilateral triangle. And the length of the shortest side is a/2, since the line we drew cut the bottom line in half.

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A. The graph shows four points on the line and the triangles resulting from sketching on the rise and run from one point to the

Gnoma [55]

Answer:

A. Yes

B. The triangles are similar

C. The corresponding sides are proportional

D. The slope of ΔABC = tanθ₁ = tanθ₂ = The slope pf ΔDEF, therefore, the slope of any two points on the line are congruent by similar triangles

E. The rate of proportionality = 2

Step-by-step explanation:

A. From the given graph, given that the angles are formed between the transversal and lines parallel to the R-value axis and the transversal and lines parallel to the thickness axis, the angle formed by the transversal and each parallel lines are corresponding angles and are therefore congruent

The tan of one of the angles on each triangle are

On the uppermost triangle, tanθ₁ = (9.42 - 3.14)/(3 - 1) = 3.14

On the other triangle, tanθ₂ =(25.12 - 12.56)/(8 - 4) = 3.14

tanθ₁ = tanθ₂

∴ θ₁ = θ₂

ΔABC and ΔDEF are right triangles, therefore, all the angles of ΔABC and ΔDEF are congruent

B. Given that the interior angles of the triangles formed by the line the graph are congruent, the triangles are similar by AA similarity postulate

C. Given that the triangles are similar, the length of the corresponding sides of the triangles are proportional

D. Given that the slope is the tangent of the angle between the line of the graph and the Thickness axis, and that the angle is the same for both triangles, (∠θ₁ and ∠θ₂) the slope or the rise-to-run ratio is constant for the two triangles and also for any two points on the line

E. The Unit of proportionality is found using any two points as follows;

Using points used to show the proportional relationship on triangle ΔABC are (8, 25.12) and (4, 12.56) and on triangle ΔDEF, the two points used to show proportional relationship are (3, 9.42), and (1, 3.14)

The ratio of the corresponding sides on ΔABC and ΔDEF are

$\overline {AC}$ / $\overline {DF}$ = (25.12 - 12.56)/(9.42 - 3.14) = 2

$\overline {BC}$ / $\overline {EF}$ = (8 - 4)/(3 - 1) = 2

Therefore, the unit rate of proportionality which is the scale factor = 2.

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