Question

find the shortest side of \\( \triangle wxy \\).

Explanation:

Step1: Sum of angles in triangle

The sum of the interior angles of a triangle is \(180^\circ\). So, \((p + 56)^\circ+(p + 54)^\circ+8p^\circ=180^\circ\).

Step2: Simplify the equation

Combine like terms: \(p + 56 + p + 54 + 8p=180\), which simplifies to \(10p+110 = 180\).

Step3: Solve for \(p\)

Subtract 110 from both sides: \(10p=180 - 110=70\). Then divide by 10: \(p = 7\).

Step4: Find each angle

- Angle at \(W\): \(p + 56=7 + 56 = 63^\circ\)
- Angle at \(X\): \(p + 54=7 + 54 = 61^\circ\)
- Angle at \(Y\): \(8p=8\times7 = 56^\circ\)

Step5: Relate angles to sides

In a triangle, the shortest side is opposite the smallest angle. The smallest angle is \(56^\circ\) (at \(Y\)), so the side opposite to \(Y\) is \(WX\). Wait, no, wait: angle at \(Y\) is \(56^\circ\), the side opposite to \(Y\) is \(WX\)? Wait, no, let's label the triangle: vertices \(W\), \(X\), \(Y\). Side opposite \(W\) is \(XY\), opposite \(X\) is \(WY\), opposite \(Y\) is \(WX\). Wait, angle at \(Y\) is \(56^\circ\), angle at \(X\) is \(61^\circ\), angle at \(W\) is \(63^\circ\). So the smallest angle is \(56^\circ\) (at \(Y\)), so the side opposite to \(Y\) is \(WX\)? Wait, no, wait the side lengths: wait, maybe I mixed up. Wait, the angles: angle \(Y = 56^\circ\), angle \(X=61^\circ\), angle \(W = 63^\circ\). So the smallest angle is \(56^\circ\) (angle \(Y\)), so the side opposite angle \(Y\) is \(WX\)? Wait, no, let's check the side labels. Wait, the side at \(Y\) is \(WY\)? Wait, no, the triangle has sides: \(WX\), \(WY\), \(XY\). Wait, angle at \(Y\) is between \(WX\) and \(XY\), so the side opposite angle \(Y\) is \(WX\). Wait, but let's recast: the side opposite the smallest angle is the shortest side. The smallest angle is \(56^\circ\) (angle \(Y\)), so the side opposite angle \(Y\) is \(WX\)? Wait, no, maybe I made a mistake. Wait, angle at \(Y\) is \(56^\circ\), so the side opposite to angle \(Y\) is \(WX\). Wait, but let's calculate the sides? Wait, no, we can also see the sides in terms of the expressions? Wait, no, the angles are \(56^\circ\) (angle \(Y\)), \(61^\circ\) (angle \(X\)), \(63^\circ\) (angle \(W\)). So the smallest angle is \(56^\circ\) (angle \(Y\)), so the side opposite angle \(Y\) is \(WX\). Wait, but let's check the side lengths. Wait, maybe the side at \(Y\) is \(WY\) with length... Wait, no, the problem is to find the shortest side. Wait, the angles: angle \(Y = 56^\circ\), angle \(X = 61^\circ\), angle \(W=63^\circ\). So the order of angles from smallest to largest: \(Y(56^\circ)<X(61^\circ)<W(63^\circ)\). Therefore, the sides opposite these angles: side opposite \(Y\) is \(WX\), side opposite \(X\) is \(WY\), side opposite \(W\) is \(XY\). So the shortest side is opposite the smallest angle, so \(WX\) is opposite \(Y\) (smallest angle), so \(WX\) is the shortest? Wait, no, wait maybe I mixed up the side labels. Wait, the triangle is \(WXY\), so vertices \(W\), \(X\), \(Y\). So side \(WX\) is between \(W\) and \(X\), side \(XY\) is between \(X\) and \(Y\), side \(WY\) is between \(W\) and \(Y\). Then angle at \(W\) is between \(WX\) and \(WY\), angle at \(X\) is between \(WX\) and \(XY\), angle at \(Y\) is between \(WY\) and \(XY\). So side opposite angle \(W\) is \(XY\), side opposite angle \(X\) is \(WY\), side opposite angle \(Y\) is \(WX\). So angle \(Y = 56^\circ\) (smallest), so side opposite \(Y\) is \(WX\), angle \(X = 61^\circ\), side opposite \(X\) is \(WY\), angle \(W = 63^\circ\), side opposite \(W\) is \(XY\). So the shortest side is \(WX\)? Wait, but let's check the lengths in terms of \(p\). Wait, no, maybe the sides are labeled with the angles? Wait, no, the angles are given in terms of \(p\), and we found \(p = 7\). Wait, maybe the side at \(Y\) is \(8p\), which is \(56\) (when \(p = 7\)), but that's an angle? Wait, no, wait the problem: the angle at \(Y\) is \(8p\) degrees? Wait, yes, the angle at \(Y\) is \(8p^\circ\), angle at \(W\) is \(p + 56^\circ\), angle at \(X\) is \(p + 54^\circ\). So we found \(p = 7\), so angle \(Y = 56^\circ\), angle \(X = 61^\circ\), angle \(W = 63^\circ\). So the sides: in a triangle, the length of a side is proportional to the sine of its opposite angle (Law of Sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)). So the side opposite the smallest angle (\(56^\circ\), angle \(Y\)) is the shortest. The side opposite angle \(Y\) is \(WX\). Wait, but maybe I made a mistake in the side - angle correspondence. Let's re - label:

- Vertex \(W\): angle \(W=(p + 56)^\circ\), side opposite is \(XY\)
- Vertex \(X\): angle \(X=(p + 54)^\circ\), side opposite is \(WY\)
- Vertex \(Y\): angle \(Y = 8p^\circ\), side opposite is \(WX\)

So angle \(Y = 56^\circ\) (smallest), so side \(WX\) (opposite \(Y\)) is the shortest? Wait, no, wait the side \(WY\) is opposite angle \(X\) (\(61^\circ\)), side \(XY\) is opposite angle \(W\) (\(63^\circ\)), side \(WX\) is opposite angle \(Y\) (\(56^\circ\)). So since \(56^\circ<61^\circ<63^\circ\), the side opposite \(56^\circ\) ( \(WX\)) is the shortest. But wait, maybe the problem has a typo, or maybe I misread the side labels. Wait, alternatively, maybe the side at \(Y\) is a side length? Wait, no, the labels: the angles are at \(W\): \(p + 56^\circ\), at \(X\): \(p + 54^\circ\), at \(Y\): \(8p^\circ\). So after finding \(p = 7\), the angles are \(63^\circ\), \(61^\circ\), \(56^\circ\). So the smallest angle is \(56^\circ\) (at \(Y\)), so the side opposite to \(Y\) is \(WX\), which is the shortest side.

Wait, but let's check again. Wait, maybe the side lengths are related to the angles. Wait, no, in a triangle, the larger the angle, the longer the side opposite to it. So the smallest angle is \(56^\circ\) (angle \(Y\)), so the side opposite angle \(Y\) (which is \(WX\)) is the shortest.

Answer:

The shortest side of \(\triangle WXY\) is \(WX\) (assuming the side opposite the \(56^\circ\) angle (at \(Y\)) is \(WX\); if we consider the side lengths in terms of the angle - side relationship, the side opposite the smallest angle is the shortest, and with \(p = 7\), the smallest angle is \(56^\circ\) (at \(Y\)), so the side opposite \(Y\) ( \(WX\)) is the shortest).