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andriy [413]
2 years ago
10

A right triangle has a 30o angle. the leg adjacent to the 30o angle measures 25 inches. what is the length of the other leg

Mathematics
1 answer:
adelina 88 [10]2 years ago
5 0
Since the triangle has a 30° angle and a right angle which is 90°, this triangle must be a 30 - 60 - 90 triangle, which is a triangle with special properties. The length of the leg adjacent to the 30° angle is equal to the √3 times x where x is the length of the angle adjacent to the 60<span>° angle. Here is a visual representation:



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This Must Only Have One Solution, Because The Right Side Of The Equation Is Just Plainly Ten. Lets Solve This:

5x + 3x - 4 = 10

Add Four To Both Sides To Begin Simplifying.

5x + 3x = 14

Now, Combine Like Terms.

8x = 14

Divide:

8x/8 = 1X = X
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14/8 = 1.75

X = 1.75
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I love these. It's often called the Shoelace Formula. It actually works for the area of any 2D polygon.


We can derive it by first imagining our triangle in the first quadrant, one vertex at the origin, one at (a,b), one at (c,d), with (0,0),(a,b),(c,d) in counterclockwise order.


Our triangle is inscribed in the a \times d rectangle. There are three right triangles in that rectangle that aren't part of our triangle. When we subtract the area of the right triangles from the area of the rectangle we're left with the area S of our triangle.


S = ad - \frac 1 2 ab -  \frac 1 2 cd - \frac 1 2 (a-c)(d-b) = \frac 1 2(2 ad - ab -cd - ad +ab +cd -bc) = \frac 1 2(ad -bc)


That's the cross product in the purest form. When we're away from the origin, a arbitrary triangle with vertices A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) will have the same area as one whose vertex C is translated to the origin.


We set a=x_1 - x_3, b= y_1  - y_3, c=x_2 - x_3, d=y_2- y_3


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That's a perfectly useful formula right there. But it's usually multiplied out:


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S= x_1 y_2 - x_2 y_1  + x_2y_3 - x_3y_2   + x_3 y_1 - x_1 y_3


That's the usual form, the sum of cross products. Let's line up our numbers to make it easier.


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