Question

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Explanation:

Step1: Find the value of \( t \)

The sum of the interior angles of a triangle is \( 180^\circ \). So we set up the equation:
\[
(t + 61) + 72t + 46t = 180
\]
Combine like terms:
\[
(1 + 72 + 46)t + 61 = 180
\]
\[
119t + 61 = 180
\]
Subtract 61 from both sides:
\[
119t = 180 - 61
\]
\[
119t = 119
\]
Divide both sides by 119:
\[
t = 1
\]

Step2: Calculate each angle

• Angle \( K \): \( t + 61 = 1 + 61 = 62^\circ \)
• Angle \( I \): \( 72t = 72\times1 = 72^\circ \)
• Angle \( J \): \( 46t = 46\times1 = 46^\circ \)

Step3: Relate angles to side lengths

In a triangle, the larger the angle, the longer the side opposite it.

• Opposite angle \( J \) (\( 46^\circ \)): side \( IK \)
• Opposite angle \( K \) (\( 62^\circ \)): side \( IJ \)
• Opposite angle \( I \) (\( 72^\circ \)): side \( JK \)

So the order from shortest to longest side is based on the order of the angles from smallest to largest (\( 46^\circ < 62^\circ < 72^\circ \)), so the sides are \( IK \), \( IJ \), \( JK \) (or in terms of the triangle's vertices, the sides opposite the angles: side opposite \( J \) is \( IK \), opposite \( K \) is \( IJ \), opposite \( I \) is \( JK \)). Wait, actually, let's label the triangle: vertices \( I \), \( J \), \( K \). So side \( IJ \) is between \( I \) and \( J \), side \( JK \) is between \( J \) and \( K \), side \( IK \) is between \( I \) and \( K \).

Angle at \( J \) is \( 46^\circ \), so side opposite ( \( IK \) ) is opposite \( 46^\circ \).

Angle at \( K \) is \( 62^\circ \), so side opposite ( \( IJ \) ) is opposite \( 62^\circ \).

Angle at \( I \) is \( 72^\circ \), so side opposite ( \( JK \) ) is opposite \( 72^\circ \).

Since \( 46^\circ < 62^\circ < 72^\circ \), the sides opposite are \( IK < IJ < JK \).

Answer:

The side lengths of \( \triangle IJK \) in order from shortest to longest are \( IK \), \( IJ \), \( JK \) (or using the angle - side relationship, the sides opposite the angles \( 46^\circ \), \( 62^\circ \), \( 72^\circ \) respectively, so the order is \( IK < IJ < JK \)).