If your referring to you maths question I haven't seen you state the answer.
Geometry determining solutions
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Consider the following diagram:
Do you see now why we have two possible triangles? Can you also see the supplementary relationship between \(R\) and \(R'\)? Please explain in your own words how you know these two angles must be supplementary.
rachelmaddie
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Supplementary angles are equal to 180 degrees
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The sum of supplementary angles is \(180^{\circ}\). I was hoping you would see that:
[MATH]\angle QR'R=\angle QRR'[/MATH]
since [MATH]\triangle QRR'[/MATH]
is isosceles. And then you would note that \(\angle QR'R\) and \(QR'P\) lie along a straight line. Thus, the two possible angles \(R\) must be supplementary.
Observe also that the altitude \(h\) of the triangle is:
[MATH]h=11\sin(37^{\circ})\approx6.619965254672531[/MATH]
Since this is less than 9, a line segment of length 9 originating at \(Q\) will have two possible points of intersection with the base of the triangle, at points \(R\) and \(R'\).
[MATH]\angle QR'R=\angle QRR'[/MATH]
since [MATH]\triangle QRR'[/MATH]
is isosceles. And then you would note that \(\angle QR'R\) and \(QR'P\) lie along a straight line. Thus, the two possible angles \(R\) must be supplementary.
Observe also that the altitude \(h\) of the triangle is:
[MATH]h=11\sin(37^{\circ})\approx6.619965254672531[/MATH]
Since this is less than 9, a line segment of length 9 originating at \(Q\) will have two possible points of intersection with the base of the triangle, at points \(R\) and \(R'\).