In triangle ABC, C is a right angle. Side c=9 and side a=5. What are the measures of the remaining sides and angles, in degrees, of the triangle? Round answers to the nearest hundredth if needed.
b^2 = 81-25
find the angles using your standard trig functions; you have all three sides.
To find the measures of the remaining sides and angles of triangle ABC, we can apply the Pythagorean Theorem and trigonometric ratios.
Given that angle C is a right angle, we can determine the lengths of sides a and b using the Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (side c) is equal to the sum of the squares of the other two sides.
Using the Pythagorean Theorem:
a^2 + b^2 = c^2
Substituting the given values:
5^2 + b^2 = 9^2
Simplifying:
25 + b^2 = 81
b^2 = 81 - 25
b^2 = 56
Taking the square root of both sides:
b ≈ √56
b ≈ 7.48
So, we have found side b to be approximately 7.48.
Next, we can determine the remaining angle of the triangle, angle A:
Since angle C is a right angle, the sum of angle A and angle B must be 90 degrees.
Angle A = 90 - Angle C
Angle A ≈ 90 - 90
Angle A ≈ 0 degrees
Finally, we can find angle B:
Angle B = 180 - Angle A - Angle C
Angle B ≈ 180 - 0 - 90
Angle B ≈ 90 degrees
To summarize, the measures of the remaining sides and angles of triangle ABC, rounded to the nearest hundredth, are as follows:
Side a ≈ 5
Side b ≈ 7.48
Side c = 9 (the hypotenuse)
Angle A ≈ 0 degrees
Angle B ≈ 90 degrees
Angle C = 90 degrees