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Sagot :
The image of the triangle is missing, so i have attached it.
Also, the options are;
A) The apothem can be found using the Pythagorean theorem.
B) The apothem can be found using the tangent ratio.
C) The perimeter of the equilateral triangle is 15 cm.
D) The length of the apothem is approximately 2.5 cm.
E) The area of the equilateral triangle is approximately 65 cm².
Answer:
Options A, B & D are true
Step-by-step explanation:
A) We are told that the triangle is an equilateral triangle.
Thus, the 3 sides equal 8.7 cm.
Now from the image, b is half of one of the sides.
Thus, b = 8.7/2 = 4.35 cm
Now, we want to find the apothem "a".
Using Pythagoras theorem, we have;
a² + 4.35² = 5²
a² = 25 - 18.9225
a² = 6.0775
a = √6.0775
a ≈ 2.5 cm.
Thus, option A is true as Pythagoras theorem was able to calculate the apothem "a"
B) since the main triangle is equilateral, it means each angle is 60°.
Now, the radius line from one corner point to the center will divide the 60° angle into 2 equal parts.
Thus, the angle made by the radius line with the base of the triangle is 60/2 = 30°
Now, from tangent ratios, we know that;
Opposite/Adjacent = tan θ
In the small triangle, opposite is apothem "a" while adjacent is b = 4.35 cm
Thus;
a/4.35 = tan 30
a = 4.35 tan 30
a ≈ 2.5 cm.
Thus, option B is correct as Tangent ratio can be used to find the apothem
C) The perimeter of equilateral triangle = 3x
Where x is length of one side.
One side is 8.7 cm
Thus, perimeter = 3 × 8.7 = 26.1 cm
Thus option C is not correct.
D) From calculations above, we saw that in both options A & B, the apothem is approximately 2.5 cm.
Thus, option D is true.
E) Area of triangle is;
A = ½ × base × height
Base = 8.7 cm
To get height, we will use trigonometric ratio. Thus;
h/8.7 = sin 60°
h = 8.7 sin 60°
Thus;
A = ½ × 8.7 × 8.7 sin 60°
A = 32.77 cm²
Thus is not equal to 65 cm². Thus, option E is not correct.
The statements that are true for the considered equilateral triangle are:
- Option A: The apothem can be found using the Pythagorean theorem.
- Option B: The apothem can be found using the tangent ratio.
- Option D: The length of the apothem is approximately 2. 5 cm.
What is apothem?
For a regular polygon, the apothem is the line segment from the center of the polygon to the mid of one of the sides of the considered polygon.
For this case, referring to the figure attached below, we have:
- Length of the apothem = x (assume)
- The triangle ABC is equilateral
- Length of line segment from vertex to center (radius) = 5 cm
- Length of each side of the triangle = 8.7 cm
Since the triangle is equilateral, there can be 6 congruent triangles like BOD.
Thus, Area of triangle ABC = 6 × Area of triangle BOD
Area of BOD = [tex]\dfrac{1}{2} \times x \times 4.35 = 2.175x \: \rm cm^2[/tex]
Area of ABC = [tex]6 \times 2.175x = 13.05x \: \rm cm^2[/tex]
The length of the apothem is x cm, and can be obtained by using Pythagoras theorem.
We can also use the tangent ratio as the line segment BO is bisecting the internal angle B (each internal angle is of 60 degrees in an equilateral triangle), so angle OBD is of 60/2 = 30 degrees. So using angle and the fact that tangent ratio is ratio of side opposite to angle and the remaining adjacent side(not hypotenuse).
[tex]tan(30^\circ) = x/4.35 \implies x = 4.35 \times \tan(30^\circ) \approx 2.5 \: \rm cm[/tex]
Using Pythagoras theorem too, we can get the value of x as:
[tex]5^2 = 4.35^2 + x^2\\x = \sqrt{25 - 18.9225} \approx 2.465 \: \rm cm \approx 2.5 cm[/tex]
Thus, area of the equilateral triangle ≈ [tex]13.05 \times x \approx 13.05 \times 2.5 = 32.575 \: \rm cm^2[/tex]
This isn't 65 sq. cm.
Thus, the statements that are true for the considered equilateral triangle are:
- Option A: The apothem can be found using the Pythagorean theorem.
- Option B: The apothem can be found using the tangent ratio.
- Option D: The length of the apothem is approximately 2. 5 cm.
Learn more about tangent ratio here:
https://brainly.com/question/14169279
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