Two trig questions (Triangles and Trapezoid)

In summary, a disabled ship was sighted 16km apart from two lighthouses. If the angle between the two lighthouses is known, the distance between the lighthouse can be calculated.
  • #1
rought
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Homework Statement



An isosceles trapezoid has a height of 4 cm and bases 3 cm and 7 cm long. How long are its diagonals?


From lighthouses P and Q, 16km apart, a disabled ship S is sighted. If angle SPQ = 44 and angle SQP = 66, find the distance from S to the nearer lighthouse?



I know how to do all of the calculations but I am just not sure where to start on these =/ ?
 
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  • #2
Well, you need to show some work or at least show your thoughts on how to proceed. What have you done so far?

Hint/question: What is nice about the trapezoid being isosceles?
 
  • #3
For #1, draw a picture of the trapezoid on a coordinate axis system. I would center the long base on the x axis. Find the coordinates of the four vertices and calculate the distance from the opposite vertices. They should be equal.

For #2, draw a picture of the triangle formed by the two lighthouses and the ship, and label the vertices as P, Q, and S (the ship). Label the side you know and the angles you know. From the given information, use what you know about trig to find the missing side. You'll probably need to use the Law of Sines.
 
  • #4
Alright well I am good with the second question on the ship i am pretty sure i got it...

For the trapezoid here's what I drew

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I'm not sure where to go from here
 
  • #5
Look a bit closer at what I said earlier:
For #1, draw a picture of the trapezoid on a coordinate axis system. I would center the long base on the x axis. Find the coordinates of the four vertices and calculate the distance from the opposite vertices. They should be equal.
 
  • #6
Alright I graphed it out, with (0,0) and (0,7) being the two bottom vertices, and (2,4) and (5,4) for the others... would the distance be 4? Between one of the long vertices and one of the short?
 
  • #7
Just use the Pythagorean theorem to calculate the diagonal (which would be the hypotenuse). You already know the height, and you can easily find out the other cathetus (you more or less already have it since you wrote out coordinates).
 
  • #8
ahh ok, √5² + 4² = √41 ≈ 6.403 that seem right?
 
  • #9
rought said:
ahh ok, √5² + 4² = √41 ≈ 6.403 that seem right?
Correct.
 
  • #10
Why 5? The right triangle with the lower right corner as one vertex and the diagonal as hypotenuse has height 4 cm, the height of the parallelogram, and base 7- 3= 4 cm.
 
  • #11
HallsofIvy said:
Why 5? The right triangle with the lower right corner as one vertex and the diagonal as hypotenuse has height 4 cm, the height of the parallelogram, and base 7- 3= 4 cm.
No, the base is 7-2=5 cm.
 

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FAQ: Two trig questions (Triangles and Trapezoid)

1. What is the Pythagorean Theorem and how is it used in triangles?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It is commonly used to find the missing side length in a right triangle, as well as to determine if a triangle is a right triangle.

2. How do you find the area of a triangle?

To find the area of a triangle, you can use the formula A = 1/2 * b * h, where b is the length of the base and h is the height of the triangle. Alternatively, you can use the fact that the area of a triangle is equal to half the product of two sides and the sine of the included angle: A = 1/2 * a * b * sin(C).

3. What is a congruent triangle?

A congruent triangle is a triangle that has the same size and shape as another triangle. This means that all three sides and all three angles are equal in both triangles. Congruent triangles can be proven through various methods, such as side-side-side (SSS), angle-angle-angle (AAA), and side-angle-side (SAS) congruence.

4. How do you find the perimeter of a trapezoid?

The perimeter of a trapezoid is equal to the sum of all its side lengths. To find the perimeter, you can add together the lengths of the top and bottom bases, as well as the lengths of the two non-parallel sides. Alternatively, you can use the formula P = a + b + c + d, where a and b are the lengths of the parallel sides and c and d are the lengths of the non-parallel sides.

5. What is the sum of the interior angles in a triangle?

The sum of the interior angles in a triangle is always 180 degrees. This is known as the Triangle Sum Theorem. This theorem can be used to find the measure of a missing angle in a triangle, as well as to determine if a given set of angles can form a valid triangle.

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