Solving Triangles
In triangle \(ABC\), \(a=10\), \(b=11\) and \(c=12\). If \(\cos A:\cos C\) can be expressed as an irreducible ratio \(m:n\), where \(m\) and \(n\) are positive integers, what is the value of \(m+n\)?
Let \(ABC\) be a triangle and let \(a\), \(b\) and \(c\) be the lengths of the sides opposite to the vertices \(A\), \(B\) and \(C\), respectively. If \( a - 3b + c = 0 \) and \( 2a + b - 2c = 0 \), the ratio \(\sin A : \sin B : \sin C\) can be expressed as \(p:q:r\), where \(p\), \(q\) and \(r\) are coprime positive integers. What is the value of \(p+q+r\)?
In a convex quadrilateral \(ABCD,\) we are given the following three angles and side length: \[\angle B=\angle D=90^\circ, \angle A=45^\circ, \lvert\overline{AC}\rvert= 3\sqrt{2}.\] What is the measure of the diagonal \(\overline{BD} ?\)
Alice and Bob start walking at the same time from the same spot towards their respective directions, forming a \(60^\circ\) angle. Their speeds are \(2\)m and \(3\)m per second, respectively. If the distance between them \(4\) seconds after the start can be expressed as \(a\sqrt{b}\), where \(b\) is prime number, what is \(a+b\)?
In triangle \(ABC\), \(\angle B=30^{\circ}\), \(\angle C=90^{\circ}\), and \(D\) is a point on side \(BC\) such that \(\overline{BD}=60\) and \(\angle ADC=45^{\circ}\). The length of side \(AC\) can be expressed as \(p(1+\sqrt{3})\). What is the value of \(p\)?