In triangle ABC, angle B is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth.
a = 3, c = 3
a^2 + c^2 = b^2 so b = 3√2
Since the sides are equal, angle A = C.
Since 180 - B = 90, A = C = half of 90° = 45°
To find the remaining sides and angles of triangle ABC, we need to use the properties of right triangles and trigonometric ratios.
Given: In triangle ABC, angle B is a right angle.
To find the remaining sides, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's label the sides:
- Side b is the side opposite angle B (the hypotenuse).
- Side a is the side adjacent to angle A.
- Side c is the side opposite angle C.
Given values:
- a = 3
- c = 3
Using the Pythagorean theorem, we can find the value of side b:
b^2 = a^2 + c^2
b^2 = 3^2 + 3^2
b^2 = 9 + 9
b^2 = 18
b ≈ √18
b ≈ 4.2 (rounded to the nearest tenth)
So, the remaining side b is approximately 4.2.
Now, let's find the angles. We can use trigonometric ratios.
For angle A, we can use the sine ratio:
sin(A) = opposite / hypotenuse
sin(A) = a / b
sin(A) = 3 / 4.2
A ≈ sin^(-1)(3 / 4.2)
A ≈ 54.7 degrees (rounded to the nearest tenth)
For angle C, we can use the cosine ratio:
cos(C) = adjacent / hypotenuse
cos(C) = c / b
cos(C) = 3 / 4.2
C ≈ cos^(-1)(3 / 4.2)
C ≈ 35.3 degrees (rounded to the nearest tenth)
Summarizing the results:
- Side a ≈ 3
- Side b ≈ 4.2
- Side c ≈ 3
- Angle A ≈ 54.7 degrees
- Angle B = 90 degrees (right angle)
- Angle C ≈ 35.3 degrees