Question

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Traduction:
Calculate the base (x) of the following figure built with 2 right triangles

Answer 1

x = 38.74

Explanation:

• We know that the base of the entire figure is made of the bases of both triangles.

Thus, we can find the bases of the two triangles and add them to find the base of the entire figure.

• Since we have two right triangles, we can find the base of each triangle using trigonometry.

Finding the base of the first triangle

• Let's call the triangle with the 60° angle and the side with a length of 30 units triangle A and let's call its base side z.

• The Triangle Sum Theorem says that the sum of a triangle's angle measures always equal 180°.
• Since we have a right triangle, the measure of the third angle in this triangle must be 30° as 180 - (60 + 90) = 30 and 60 + 90 + 30 = 180.

• A triangle with a 30°, 60°, and 90° (right) angle is called a 30-60-90 triangle.

The sides of 30-60-90 triangles adhere to the following rules:

• The side opposite the 30° angle is the shortest side and its length can be referred to as a.
• The side opposite the 60° angle is the medium/longer side and its length can be referred to as a*√(3).
• The side opposite the 90° (right) angle is the longest side called the hypotenuse and its length can be referred to as 2a.

Since the side opposite the 60° angle is 30, we can find a by applying the rule for the side opposite the 60° angle in a 30-60-90 triangle:

Step 1: Make an equation out of the fact that the side opposite the 60° equals a*√(3)

30 = a*√(3)

Step 2: Divide both sides by √(3) to find a, the length of the base of triangle A:

(30 = a * √(3)) / √(3)

30 / √(3) = a

17.32050808 = a

• It's better not to round since rounding now at this intermediate step may interfere with our work when trying to find the base of the entire figure.

Thus, the length of the base of triangle A is 17.32050808 units.

Finding the base of the second triangle:

• We can call the triangle with the 35° angle and the side with a length of 15 units triangle B.
• We can allow z to represent the length of the base of triangle B.
• Since we have a right triangle, we'll need to use one of the trigonometric ratios to find z, the length of the base of triangle B.
• When the 35° angle is our reference angle, z is the adjacent side and the 15 unit side is the opposite side.
• Thus, we can use the tangent ratio to find z, the length of triangle B's base.
• The tangent ratio is given by:

tan(θ) = opposite / adjacent, where

• θ is the reference angle

Now we can plug in 35 for θ and 15 for the opposite side to find the adjacent side (aka the length of the base of triangle B):

Step 1: Plug in 35 for θ and 15 for the opposite side:

tan(35) = 15 / adjacent

Step 2: Multiply both sides by the adjacent side:

(tan(35) = 15 / adjajcent) * adjacent

adjacent * tan(35) = 15

Step 3: Divide both sides by tan(35) to find the length of the adjacent side (i.e., the length of the base of triangle B):

(adjacent * tan(35) = 15) / tan(35)

adjacent = 15 / tan(35)

adjacent = 21.4222201

Thus, the length of the base of triangle B is 21.4222201 units.

Find the base(x) of the figure:

Now we can add the lengths of the bases of triangles A and B to find x, the base of the figure:

Length of triangle A's base + Length of triangle B's base = base of figure (x)

17.32050808 + 21.4222201 = x

38.74272818 = x

38.74 = x

Thus, the base of the figure built with 2 right triangles is 38.74.