there are two legs on a triagle on is 20 the othe is 30 what is the missing angle

Answer 1

To find the missing angle in a triangle where you know two sides, you can use the Law of Cosines. In your case, you have a triangle with two sides of lengths \(a = 20\) and \(b = 30\).

Assuming these two sides are opposite to angles A and B respectively, we will denote the third side (the side opposite the missing angle C) as \(c\). However, to find the missing angle without additional information (like the length of the third side or the included angle), we generally cannot determine the angles of a triangle uniquely.

If you need to find angle \(C\) specifically, you would need the length of side \(c\). However, if you're trying to find the angle opposite to one of the known sides, you would need some additional information.

If you know that this triangle is a right triangle, then:

1. If 20 and 30 are the two legs of a right triangle, you can use the Pythagorean theorem to find the hypotenuse \(c\): \[ c = \sqrt{20^2 + 30^2} = \sqrt{400 + 900} = \sqrt{1300} \approx 36.06. \]

2. Then you can use trigonometric functions to find the angles. For example, you can find angle \(A\) (opposite side \(a = 20\)) using the sine function: \[ \sin(A) = \frac{a}{c} = \frac{20}{36.06} \approx 0.555. \] \[ A \approx \sin^{-1}(0.555) \approx 33.73^\circ. \]

3. The other angle \(B\) can be found similarly: \[ \sin(B) = \frac{b}{c} = \frac{30}{36.06} \approx 0.831. \] \[ B \approx \sin^{-1}(0.831) \approx 56.27^\circ. \]

4. Finally, angle \(C\) can be found using: \[ C = 180^\circ - A - B \approx 180^\circ - 33.73^\circ - 56.27^\circ \approx 90^\circ. \]

If you have any other constraints or information about the triangle, please provide them so I can assist you further!

Answer 2

If a right triangle has legs measuring 10 cm and 24 cm, how long is the hypotenuse?

Answer 3

To find the length of the hypotenuse in a right triangle when you know the lengths of the two legs, you can use the Pythagorean theorem, which states:

\[ c^2 = a^2 + b^2 \]

where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the legs.

In your case:

• \(a = 10\) cm
• \(b = 24\) cm

Now, plug the values into the equation:

\[ c^2 = 10^2 + 24^2 \] \[ c^2 = 100 + 576 \] \[ c^2 = 676 \]

Now, take the square root of both sides to find \(c\):

\[ c = \sqrt{676} = 26 \text{ cm} \]

Therefore, the length of the hypotenuse is 26 cm.

Answer 4

Camila needs to fix the window on the front of her house. The bottom of the window is 16 ft. from the ground. She only has a 20 ft. ladder. How far from the base of the house should she set up the ladder so that the top of the ladder is exactly at the bottom of the window

Answer 5

To determine how far from the base of the house Camila should set up the ladder, we can use the Pythagorean theorem. In this scenario, the ladder acts as the hypotenuse of a right triangle, with one leg being the height of the window from the ground and the other leg being the distance from the base of the house to the base of the ladder.

Given:

• Height of the window (one leg of the triangle) = 16 ft
• Length of the ladder (hypotenuse) = 20 ft

Let \(d\) be the distance from the base of the house to the base of the ladder. According to the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]

where:

• \(a\) is the height of the window (16 ft),
• \(b\) is the distance from the base of the house (the unknown \(d\)),
• \(c\) is the length of the ladder (20 ft).

Plugging in the values into the formula:

\[ 16^2 + d^2 = 20^2 \] \[ 256 + d^2 = 400 \]

Now, isolate \(d^2\):

\[ d^2 = 400 - 256 \] \[ d^2 = 144 \]

Now, take the square root of both sides to find \(d\):

\[ d = \sqrt{144} = 12 \]

Therefore, Camila should set up the ladder 12 ft from the base of the house.

Answer 6

Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41st Street and Edison Avenue. Her house is at the corner of 42nd Street and Johnson Avenue. She can walk down Johnson Avenue and turn right on 41st Street, or she can walk through the park. The distance from 42nd Street to 41st Street is 80 yards, and it is 60 yards from Johnson Avenue to Edison Avenue. Assuming the street intersections are 90°
angles, how much shorter is it for Sylvia to walk through the park

Answer 7

To determine how much shorter it is for Sylvia to walk through the park compared to walking along the streets, we can calculate the distances for both routes.

1. Distance Walking Along the Streets:

   • Sylvia needs to walk down Johnson Avenue to reach 41st Street (80 yards), and then walk along 41st Street to get to Edison Avenue (60 yards).
   • The total distance when walking along the streets is: \[ \text{Distance along streets} = 80 \text{ yards} + 60 \text{ yards} = 140 \text{ yards} \]

2. Distance Walking Through the Park:

   • Walking through the park would form a right triangle with the two legs being the distances from 42nd Street to 41st Street (80 yards) and from Johnson Avenue to Edison Avenue (60 yards). We can use the Pythagorean theorem to find the distance: \[ d = \sqrt{(80^2) + (60^2)} \] \[ d = \sqrt{6400 + 3600} \] \[ d = \sqrt{10000} = 100 \text{ yards} \]

3. Calculating the Difference:

   • Now, we can find the difference between the two distances: \[ \text{Difference} = \text{Distance along streets} - \text{Distance through the park} \] \[ \text{Difference} = 140 \text{ yards} - 100 \text{ yards} = 40 \text{ yards} \]

Therefore, walking through the park is 40 yards shorter for Sylvia compared to walking along the streets.