Question

1. if \\( \overline { q n } \\) bisects \\( \angle p q r \\) and \\( n \\) is the midpoint of \\( \overline { p r } \\) classify each triangle by its angles and sides.

a. \\( \triangle p q r \\) :________________________
b. \\( \triangle p r t \\) :________________________
c. \\( \triangle t q s \\) :________________________
d. \\( \triangle q n p \\) :________________________

Explanation:

Step1: Classify triangle by angles and sides

• For \(\triangle PQR\):
• Angles: One angle is \(30^{\circ}+28^{\circ} = 58^{\circ}\), another is \(60^{\circ}\), and the third angle is \(180^{\circ}-(58^{\circ}+60^{\circ})=62^{\circ}\). All angles are less than \(90^{\circ}\), so it is an acute triangle.
• Sides: Since \(N\) is the mid - point of \(\overline{PR}\) and \(QN\) is an angle bisector, and there is no information about side lengths being equal, but we can assume from the given data (if we consider the general triangle classification rules based on side - angle relations). Wait, no, actually, since \(QN\) is an angle bisector and \(N\) is the mid - point of \(PR\), and if we assume the triangle has no equal sides (as no side lengths are marked as equal except in some sub - triangles). But wait, no, for \(\triangle PQR\), all angles are acute (\(30 + 28=58\), \(60\), \(180-(58 + 60)=62\)), and no sides are marked equal.
• For \(\triangle PRT\):
• Angles: There is no information about angles in \(\triangle PRT\) (assuming it's a typo, maybe \(\triangle QRT\) or another, but if we go by the problem as is. Wait, no, looking at the figure, for \(\triangle QNP\):
• Angles: One angle is \(90^{\circ}\), so it is a right triangle.
• Sides: Sides \(PN = 7\), \(QN\) (using trigonometry in \(\triangle PQN\), \(\cos30^{\circ}=\frac{QN}{14}\), \(QN = 7\sqrt{3}\approx12.12\), \(PQ = 14\)). No two sides are equal.
• For \(\triangle TQS\):
• Angles: Two angles are \(76^{\circ}\), so the third angle is \(180-(76 + 76)=28^{\circ}\). Since two angles are equal, two sides are equal.
• Sides: Two angles are \(76^{\circ}\), so it is an isosceles triangle. And all angles are less than \(90^{\circ}\) (acute).
• For \(\triangle PQR\):
• Re - checking: \(\angle PQR=30 + 28=58^{\circ}\), \(\angle QPR\): Using \(\triangle PQN\), \(\sin30^{\circ}=\frac{PN}{PQ}\), \(PN = 7\), \(PQ = 14\). In \(\triangle PQR\), using the law of sines \(\frac{PQ}{\sin60^{\circ}}=\frac{PR}{\sin58^{\circ}}=\frac{QR}{\sin62^{\circ}}\). All angles \(<90^{\circ}\) (acute), no sides equal (scalene).
• For \(\triangle QNP\):
• Angle: One angle is \(90^{\circ}\) (right - angled), sides: \(PN = 7\), \(QN\approx12.12\), \(PQ = 14\) (no two sides equal, scalene).

Answer:

a. Acute scalene triangle
b. (Assuming it's a mis - label, if we consider the problem as is, but if we go by the figure and correct labeling, maybe it's a non - existent or mis - written triangle. But if we assume the intended triangles):
a. Acute scalene triangle; b. (No valid \(\triangle PRT\) based on the figure, but if we consider \(\triangle QNP\) as part of the answer set); c. Acute isosceles triangle; d. Right scalene triangle.

Wait, re - checking:

• a. \(\triangle PQR\):
• Angles: \(\angle Q = 30+28 = 58^{\circ}\), \(\angle R=60^{\circ}\), \(\angle P=180-(58 + 60)=62^{\circ}\) (all \(<90^{\circ}\), acute). Sides: no two sides marked equal (scalene).
• **b. (Assuming it's a typo, maybe \(\triangle QRT\) is not in the question. But if we go by the problem as written, maybe it's a non - relevant triangle. But if we consider the answer for the given options in the problem's context, perhaps the user made a typo. But focusing on the given labels \(a - d\)):
• c. \(\triangle TQS\):
• Angles: \(\angle S = 76^{\circ}\), \(\angle T = 76^{\circ}\), \(\angle Q=180-(76 + 76)=28^{\circ}\) (acute, two angles equal \(\Rightarrow\) isosceles).
• d. \(\triangle QNP\):
• Angle: \(90^{\circ}\) (right), sides: \(PN = 7\), \(QN\) (calculated as above \(

eq7\), \(PQ = 14
eq7\) and \(QN
eq14\)) (scalene).

So:
a. Acute scalene triangle
b. (If we assume it's a non - existent or mis - labeled triangle in the problem's original source. But if we follow the problem's numbering \(a - d\) and the figure, and assume \(\triangle PRT\) is a mistake. But for the sake of answering as per the problem's structure):
a. Acute scalene triangle; b. (No valid \(\triangle PRT\) in the figure context for classification as per given data. But if we consider the problem's intention, maybe it's a typo. However, based on the given figure and standard triangle classification):
a. Acute scalene; c. Acute isosceles; d. Right scalene.

Final Answer
a. Acute scalene triangle
b. (No valid \(\triangle PRT\) for classification with given data. If it's a typo, but as per problem structure)
c. Acute isosceles triangle
d. Right scalene triangle