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If Mt. Pinatubo, Mayon volcano, and Taal volcano form an oblique triangle respectively. Which will generate the largest angle: from Mt. Pinatubo, from Mayon volcano, or from Taal volcano? Given that Mt. Pinatubo is 416 km away from Mayon volcano, Mayon volcano is 307 km away from Taal volcano, and Taal volcano is 144 km away from Mt. Pinatubo. Elaborate/Explain to support your answer.

User Peutch
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Answer:

Explanation:

To determine which volcano, Mt. Pinatubo, Mayon volcano, or Taal volcano, generates the largest angle in the oblique triangle, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. In this case, we want to find the largest angle, so we'll use the Law of Cosines to find the largest included angle.

Let's label the vertices of the triangle as follows:

- A: Mt. Pinatubo

- B: Mayon volcano

- C: Taal volcano

Given the distances between the volcanoes:

- AC = 144 km (distance from Mt. Pinatubo to Taal)

- AB = 416 km (distance from Mt. Pinatubo to Mayon)

- BC = 307 km (distance from Mayon to Taal)

We want to find the largest angle, which is opposite the largest side (BC) according to the Law of Cosines:

Cos(C) = (AB² + AC² - BC²) / (2 * AB * AC)

Now, we'll calculate the angles at each vertex.

1. Angle at A (Mt. Pinatubo):

Cos(A) = (BC² + AC² - AB²) / (2 * BC * AC)

2. Angle at B (Mayon volcano):

Cos(B) = (AC² + AB² - BC²) / (2 * AC * AB)

3. Angle at C (Taal volcano):

Cos(C) = (AB² + BC² - AC²) / (2 * AB * BC)

Now, let's calculate these cosines:

Angle at A (Mt. Pinatubo):

Cos(A) = (307² + 144² - 416²) / (2 * 307 * 144) ≈ 0.983

Angle at B (Mayon volcano):

Cos(B) = (144² + 416² - 307²) / (2 * 144 * 416) ≈ 0.934

Angle at C (Taal volcano):

Cos(C) = (416² + 307² - 144²) / (2 * 416 * 307) ≈ 0.916

Now, to find the largest angle, we need to find which of these cosines is the smallest (as cosines are inversely related to angles).

The largest angle will be at the vertex corresponding to the smallest cosine. So, the largest angle will be at Taal volcano (vertex C), as Cos(C) is the smallest among the three values.

Therefore, Taal volcano generates the largest angle in the oblique triangle formed by Mt. Pinatubo, Mayon volcano, and Taal volcano.

User CHARAFI Saad
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