a. What is the length of the hypotenuse of triangle ABC?

b. What is the length of the shorter leg of triangle ABC?

c. What is the length of the longer leg of triangle ABC?

angle A=60, angle B=30, angle C=90

the triangle has a height of 10

a. c = h/sinA = 10 / sin60 =

b. tanA = h/b,
tan60 = 10 / b, Solve for b.

c. a = csin60 =

To find the lengths of the sides of triangle ABC, we will use the given information and trigonometric functions.

a. To find the length of the hypotenuse (side AC), we can use the sine function with angle A. Since we know the height of the triangle is 10, we can write:

sin(A) = opposite/hypotenuse
sin(60°) = 10/hypotenuse

Using the sin(60°) = √3/2, we can solve for the hypotenuse:

√3/2 = 10/hypotenuse
hypotenuse = 10/(√3/2)
hypotenuse = 10 * (2/√3)
hypotenuse = (20/√3)

Therefore, the length of the hypotenuse (side AC) is approximately 11.55.

b. To find the length of the shorter leg (side AB), we can use the sine function with angle B. We can write:

sin(B) = 10/hypotenuse
sin(30°) = 10/AB

Using the sin(30°) = 1/2, we can solve for the shorter leg:

1/2 = 10/AB
AB = 10 * 2
AB = 20

Therefore, the length of the shorter leg (side AB) is 20.

c. To find the length of the longer leg (side BC), we can use the Pythagorean theorem. Since we know the lengths of the other two sides, we can write:

BC^2 = AC^2 - AB^2
BC^2 = (20/√3)^2 - 20^2
BC^2 = (400/3) - 400/1
BC^2 = (400 - 1200) / 3
BC^2 = -800/3

Since the resulting value is negative, it means that the triangle does not satisfy the conditions of the Pythagorean theorem. Therefore, we cannot find the length of the longer leg (side BC) with the given information.

To find the lengths of the sides of the triangle, we can use trigonometric ratios such as sine, cosine, and tangent. Given that we have the angle measures and the height of the triangle, let's proceed with the calculations:

a. The length of the hypotenuse of triangle ABC can be found using the sine ratio. In a right-angled triangle, the sine of an acute angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

sin(A) = opposite/hypotenuse

Since we know the height opposite angle A is 10, and angle A is 60 degrees, we can substitute these values into the equation:

sin(60) = 10/hypotenuse

To solve for the hypotenuse, we can rearrange the equation:

hypotenuse = 10 / sin(60)

Using a scientific calculator to find the sine of 60 (which is approximately 0.866), we can calculate:

hypotenuse = 10 / 0.866 ≈ 11.547

Therefore, the length of the hypotenuse of triangle ABC is approximately 11.547.

b. The length of the shorter leg of triangle ABC can be found using the cosine ratio. In a right-angled triangle, the cosine of an acute angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

cos(A) = adjacent/hypotenuse

Since angle A is 60 degrees and we already know the hypotenuse is approximately 11.547, we can substitute these values into the equation:

cos(60) = adjacent/11.547

Similar to before, rearrange the equation to solve for the adjacent side:

adjacent = 11.547 * cos(60)

Using a scientific calculator to find the cosine of 60 (which is approximately 0.5), we can calculate:

adjacent = 11.547 * 0.5 = 5.774

Therefore, the length of the shorter leg of triangle ABC is approximately 5.774.

c. The length of the longer leg of triangle ABC can be found by subtracting the length of the shorter leg from the height of the triangle. Given that the height is 10 and the length of the shorter leg is approximately 5.774, we can calculate:

longer leg = height - shorter leg

longer leg = 10 - 5.774 ≈ 4.226

Therefore, the length of the longer leg of triangle ABC is approximately 4.226.

To summarize:
a. The length of the hypotenuse of triangle ABC is approximately 11.547.
b. The length of the shorter leg of triangle ABC is approximately 5.774.
c. The length of the longer leg of triangle ABC is approximately 4.226.