Question

what is the smallest integer value for ( ik ) for which parallelogram ( ijkl ) will have an obtuse angle at ( j )?
a. 12
b. 13
c. 14
d. 15

Explanation:

Step1: Recall properties of parallelograms and right triangles

In a parallelogram \(IJKL\), \(IJ = 12\) and \(JK = 5\). If we consider triangle \(IJK\), when \(\angle J\) is a right angle, by the Pythagorean theorem, \(IK=\sqrt{IJ^{2}+JK^{2}}\).

Step2: Calculate \(IK\) for right angle at \(J\)

Using the Pythagorean theorem: \(IK=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\).

Step3: Determine when \(\angle J\) is obtuse

In a triangle, if the square of one side is greater than the sum of the squares of the other two sides, the angle opposite that side is obtuse. For \(\angle J\) to be obtuse in \(\triangle IJK\), \(IK^{2}>IJ^{2}+JK^{2}\). We know \(IJ = 12\), \(JK = 5\), so \(IJ^{2}+JK^{2}=144 + 25 = 169\). We need \(IK^{2}>169\), so \(IK>13\) (since \(IK\) is a length, it's positive). The smallest integer value of \(IK\) satisfying this is \(14\).

Answer:

C. 14