A TRIANGLE IS INSCRIBED IN A CIRCLE SUCH THAT ONE OF ITS SIDES IS A DIAMETER OF THE CIRCLE.IF THE RADIUS OF THE CIRCLE IS 30.5 CM AND ONE OF THE SIDES IS 60 CM, FIND THE AREA OF THE TRIANGLE?

A triangle on a diameter makes a right angle at the circle

So let the third side be x
x^2 + 60^2 = 61^2
x = √121 = 11

so the area = (1/2)(60)(11) = 330

To find the area of the triangle, we need to know the length of the altitude (perpendicular height) from the base of the triangle to the opposite vertex.

In this case, the side of the triangle that is a diameter of the circle is 60 cm. Since a diameter of a circle is twice the radius, we can find the height of the triangle by using the Pythagorean theorem.

Let's label the points of the triangle A, B, and C. Point A is the vertex opposite to the diameter (side) of length 60 cm. Point B and C are the ends of the diameter.

The radius of the circle is 30.5 cm, which is also the distance from the center of the circle to any point on the circle. Therefore, the distance from the center of the circle to point A is also 30.5 cm.

Now, we can draw a line segment from the center of the circle to point A, creating a right-angled triangle. Let's label the point where this line segment intersects the diameter as point D.

Triangle ADB is a right-angled triangle, and AD is the height (altitude) of the triangle that we need to find.

Using the Pythagorean theorem, we have:

AD^2 + BD^2 = AB^2

Since AB is the diameter and is equal to 60 cm, BD is half of AB, i.e., 30 cm.

AD^2 + (30 cm)^2 = (60 cm)^2

Simplifying further, we have:

AD^2 + 900 cm^2 = 3600 cm^2

AD^2 = 3600 cm^2 - 900 cm^2

AD^2 = 2700 cm^2

To find AD, we take the square root of both sides:

AD = √2700 cm^2

AD ≈ 51.96 cm (rounded to two decimal places)

Now, we can calculate the area of the triangle using the formula:

Area = (1/2) * base * height

In this case, the base is the side of the triangle that is a diameter (60 cm), and the height is the altitude (51.96 cm) we just found.

Area = (1/2) * 60 cm * 51.96 cm

Area ≈ 1558.8 cm^2 (rounded to one decimal place)

Therefore, the area of the triangle is approximately 1558.8 cm^2.

To find the area of the triangle, we can use the formula for the area of a triangle: A = (1/2) * base * height.

In this case, the base of the triangle is the side that is a diameter of the circle, which is 60 cm.

To find the height of the triangle, we need to consider that the height is perpendicular to the base. Since the base is a diameter of the circle, it passes through the center of the circle. This means that the height is equal to the radius of the circle, which is 30.5 cm.

Now we have all the information we need to calculate the area:

A = (1/2) * base * height
= (1/2) * 60 cm * 30.5 cm
= 30 cm * 30.5 cm
= 915 cm²

Therefore, the area of the triangle is 915 cm².