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Chapter 6

Test Yourself

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03:15

Geometry A parallelogram h…

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A rectangle has a diagonal…

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A triangle has a $45^{\cir…

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Find the sine of the small…

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Okay, This one says if the length of a rectangle is three times, it's with what is the sine of the angle that the diagonal mix with the longer side. Let's go out and start out by making a visual representation. Okay, so here's our rectangle and we know that the length over here is three times the width, so we can say three w. So then we're gonna go out and draw that diagonal from the So here's a diagonal and this is the opposite side. So we're gonna put Fada right here again because it's a rectangle. You know that this new triangle that we've made is a right triangle. So then we want to find the sign of that angle. So sign we know is actually going to be opposite overhyped pot noose because of Socotra. So okay, to, uh, so we know that it's going to be the opposite over the hypotenuse and from faith to the opposite angle is going to be w. And then the high part News is always the longest side. So this one is the high pot news. So we don't currently know the hypotenuse, but we can figure it out using Pythagorean Theorem Zachary in Durham, which says that a squared plus B squared is equal to C squared where C has to be the hypothesis. So then we can say W squared us. Www squared is equal to C square. And if you simplify this, we've got w squared was nine w squared is equal to C squared and then 10 w squared is equal to C squared. Then if we take the square root of both sides, we can isolate that variable c So now we can have C is equal to the square root of 10 times Tell you so we're gonna put here square grid of 10 times w. So then we just have to find the sign again. It's going to be the opposite over the hypotenuse so we can say sine of theta is going to be w which is the opposite divided by the hypothalamus, which is return times w So then, from here, w on the top and the w on the bottom are going to cancel out. We're going to be left with one over the square root of 10. And if we calculate that out, that's going to be 0.316 So a is going to be our answer here, and we know that works because we took the sign of the diagonal that it makes with the longer side right here.

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