right triangle ABC has area 32√3cm^2. The measure of <A = 30, m<B=90. What is the length of BC? AB? AC? Express all answer
s in simplest radical form.
1 answer:
<h3>Answers:</h3><h3>BC = 8</h3><h3>AB = 8*sqrt(3)</h3><h3>AC = 16</h3>
"sqrt" stands for "square root"
=======================================================
Explanation:
Angle C is 60 degrees because adding the angles gets A+B+C = 180. Put another way, C = 180-A-B. Therefore, C = 180-30-90 = 60.
We have a 30-60-90 triangle with
- BC as the short leg (since it is opposite the smallest angle A = 30)
- AB as the long leg (because it is opposite the medium angle C = 60)
- AC as the hypotenuse (it is opposite the largest angle, so AC is the longest side)
Check out the diagram below.
----------
From here, we use the 30-60-90 triangle template. The short leg is x, the hypotenuse is 2x and the long leg is x*sqrt(3). The area of this template triangle is
area = 0.5*base*height
area = 0.5*(short leg)*(long leg)
area = 0.5*x*(x*sqrt(3))
area = 0.5*x^2*sqrt(3)
set this equal to 32*sqrt(3) and you should find that 0.5x^2*sqrt(3) = 32*sqrt(3) solves to x = 8
----------
Therefore we can say,
- BC = short leg = x = 8
- AB = long leg = x*sqrt(3) = 8*sqrt(3)
- AC = hypotenuse = 2x = 16
You might be interested in
Explanation:
When the inequality symbol is replaced by an equal sign, the resulting linear equation is the boundary of the solution space of the inequality. Whether that boundary is included in the solution region or not depends on the inequality symbol.
The boundary line is included if the symbol includes the "or equal to" condition (≤ or ≥). An included boundary line is graphed as a solid line.
When the inequality symbol does not include the "or equal to" condition (< or >), the boundary line is not included in the solution space, and it is graphed as a dashed line.
Once the boundary line is graphed, the half-plane that makes up the solution space is shaded. The shaded half-plane will be to the right or above the boundary line if the inequality can be structured to be of one of these forms:
- x > ... or x ≥ ... ⇒ shading is to the right of the boundary
- y > ... or y ≥ ... ⇒ shading is above the boundary
Otherwise, the shaded solution space will be below or to the left of the boundary line.
_____
Just as a system of linear equations may have no solution, so that may be the case for inequalities. If the boundary lines are parallel and the solution spaces do not overlap, then there is no solution.
_____
The attached graph shows an example of graphed inequalities. The solutions for this system are in the doubly-shaded area to the left of the point where the lines intersect. We have purposely shown both kinds of inequalities (one "or equal to" and one not) with shading both above and below the boundary lines.
