Triangle RST, RS = 30 centimeters, measure of angle T = 105, and measure of angle R = 40. Find the area of triangle RST, to the nearest square centimeter.

Question

Triangle RST, RS = 30 centimeters, measure of angle T = 105, and measure of angle R = 40. Find the area of triangle RST, to the nearest square centimeter.

Answer 1

Angle S is 35 degrees. (180-105-40)

Side ST's length, which I will call r, can be obtained from the law of sines.
30/sin 105 = r/sin 40 = 31.058
Therefore r = 19.96

The height measured from base RS is:
19.96 sin 35 = 11.45

Tha triangle area is
(1/2)(base)(height) = (0.5)*(30)*11.45
= 171.8 cm^2

Answer 2

To find the area of triangle RST, we can use the formula:

Area = (1/2) * side1 * side2 * sin(angle)

Given that RS = 30 centimeters, angle T = 105°, and angle R = 40°, we need to find the length of side ST.

Using the Law of Sines, we have:

sin(angle T) / side ST = sin(angle R) / side RS

sin(105) / ST = sin(40) / 30

ST = (30 * sin(105)) / sin(40)

ST ≈ 47.88 centimeters (rounded to two decimal places)

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

Area = (1/2) * side1 * side2 * sin(angle)

Area = (1/2) * 30 * 47.88 * sin(105)

Area ≈ 664.92 square centimeters (rounded to the nearest whole number)

Therefore, the area of triangle RST is approximately 665 square centimeters.

Answer 3

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

In this case, we have the lengths of two sides, RS and ST, but we don't have the height. However, we can use the information given about the angles to find the height.

Since the sum of the angles in a triangle is always 180 degrees, we can find the measure of angle S by subtracting the measures of angles T and R from 180 degrees:
angle S = 180 - (angle T + angle R)
= 180 - (105 + 40)
= 180 - 145
= 35 degrees

Now, to find the height, we can draw an altitude from angle S to side RT, creating a right triangle. The height will be the length of this altitude.

Since angle S is opposite side RT, the height is the length of side RS multiplied by the sine of angle S:
height = RS * sin(S)
= 30 * sin(35)

Using a calculator, we find that sin(35) ≈ 0.5736.

Substituting this into the equation, we have:
height ≈ 30 * 0.5736
≈ 17.208 centimeters

Now we have the base (RS) and the height (17.208 cm), so we can calculate the area of triangle RST:
area = 1/2 * base * height
= 1/2 * 30 * 17.208
≈ 258.12 square centimeters

Therefore, the area of triangle RST is approximately 258.12 square centimeters.