Find the area of triangle whose each side is twice the side of the given triangle.?

Question

Answer

Finding Area of a Triangle

Given a triangle with sides a, b, and c, the area can be found using the following formula:

Area = (1/2) * base * height

where base is any one of the sides and height is the perpendicular distance from that side to the opposite vertex.

Problem Statement

Find the area of a triangle whose each side is twice the side of the given triangle.

Solution

Let the given triangle have sides a, b, and c. We need to find the area of a triangle whose each side is twice the side of the given triangle. Let the sides of this new triangle be 2a, 2b, and 2c.

Step 1: Finding the Base and Height of the Given Triangle

Let's assume side a is the base of the given triangle. To find the height, we can use the Pythagorean theorem:

c^2 = a^2 + b^2

Solving for b, we get:

b = sqrt(c^2 - a^2)

The height of the triangle can be found using the formula:

height = b * sin(A)

where A is the angle opposite to side a. We can use the law of cosines to find this angle:

a^2 = b^2 + c^2 - 2bc cos(A)

Solving for cos(A), we get:

cos(A) = (b^2 + c^2 - a^2) / 2bc

Now, we can find sin(A) using the identity:

sin(A) = sqrt(1 - cos^2(A))

Substituting the values, we get:

sin(A) = sqrt(1 - ((b^2 + c^2 - a^2) / 2bc)^2)

Using these values, we can find the area of the given triangle:

Area = (1/2) * a * b * sin(A)

Step 2: Finding the Base and Height of the New Triangle

Since each side of the new triangle is twice the side of the given triangle, the base of the new triangle is 2a. To find the height, we can use the same formula:

height = b * sin(A)

But since the angles of the triangle remain the same, the value of sin(A) will also remain the same. Therefore, the height of the new triangle will be:

height' = b * sin(A) = 2b * sin(A)

Step 3: Finding the Area of the New Triangle

Using the formula for the area of

Answer

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