Question

1. 15in 20in 18in 13. 14. 5ft 97° 5ft if \\( \overline{pr} \\) bisects \\( \angle srt \\) and \\( u \\) is the midpoint of \\( \overline{rt} \\), classify each triangle by its angles and sides. 15. \\( \triangle uqr \\) 16. \\( \triangle rst \\) 17. \\( \triangle srq \\) 18. \\( \triangle prt \\) 19. \\( \triangle tqu \\)

Explanation:

Step1: Analyze Triangle 12 (15in, 18in, 20in)

Check side lengths: all sides are different (15, 18, 20), so scalene. Check angles: use Pythagorean theorem. \(15^2 + 18^2 = 225 + 324 = 549\), \(20^2 = 400\). \(549 eq400\), so no right angle. All angles acute? \(20^2 < 15^2 + 18^2\) (400 < 549), so acute scalene.

Step2: Analyze Triangle 13 (Right Triangle)

Has a right angle (90°), so right triangle. Sides: legs (unknown) and hypotenuse. Since it's right-angled, classify as right (by angle) and scalene (if legs different, but diagram shows one right angle, so right triangle, scalene if legs not equal, or isoceles if legs equal. From diagram, legs look different, so right scalene.

Step3: Analyze Triangle 14 (5ft, 5ft, 97°)

Two sides equal (5,5), so isoceles. Angle 97° is obtuse (greater than 90°), so obtuse isoceles.

Step4: Analyze Triangle 15 (\(\triangle UQR\))

From diagram, \(UQ = QR\)? Wait, \(U\) is midpoint, \(PR\) bisects angle. Wait, \(\triangle UQR\): angles? \( \angle UQR = 90^\circ\) (right angle), \(UQ = QR\)? Wait, no, \(U\) is midpoint, \(RT = 8 + 8 = 16\), so \(RU = UT = 8\). \(QR = 8\), \(UQ\)? Wait, \(\triangle UQR\): right angle at \(Q\)? Wait, diagram shows right angle at \(Q\) for some triangles. Wait, \(\triangle UQR\): angle at \(Q\) is 90°? Wait, no, \(\triangle UQR\): angles: 30°, 60°, 90°? Wait, \( \angle URQ = 30^\circ\), \( \angle UQR = 90^\circ\) (right angle), so right triangle. Sides: \(RU = 8\), \(QR = 8\)? No, \(QR = 8\), \(UQ\): in 30-60-90 triangle, sides are \(x\), \(x\sqrt{3}\), \(2x\). If \(RU = 8\) (hypotenuse), then \(UQ = 4\), \(QR = 4\sqrt{3}\). Wait, maybe better: \(\triangle UQR\) has a right angle? Wait, the diagram for \(\triangle UQR\): angle at \(Q\) is 90°? No, \(\triangle UQR\): angle at \(U\) is 120°? Wait, no, earlier triangle: \(\triangle UQR\): angles: 30°, 60°, 90°? Wait, maybe I misread. Let's re-express:

For \(\triangle UQR\):
- Sides: \(UQ\) (let's see), \(QR = 8\), \(RU = 8\)? No, \(RU = 8\) (since \(U\) is midpoint, \(RT = 16\), so \(RU = 8\)). \(QR = 8\), so \(RU = QR = 8\), angle at \(R\) is 30°? Wait, no, \(\triangle UQR\): angle at \(U\) is 120°? Wait, the diagram shows \(\triangle UQR\) with angle 120°? No, the small triangle with 120° is \(\triangle UQ...\) Wait, maybe the problem is to classify each triangle (12 - 19). Let's focus on triangle 12,13,14 as per diagram.

Wait, the user's diagram has triangles 12 (15,18,20), 13 (right), 14 (5,5,97°), and triangles 15-19. Let's take triangle 12 first:

Triangle 12 (15in, 18in, 20in)

- By sides: Scalene (all sides different: 15, 18, 20)
- By angles: Acute (since \(c^2 < a^2 + b^2\) where \(c = 20\), \(a = 15\), \(b = 18\); \(20^2 = 400\), \(15^2 + 18^2 = 225 + 324 = 549\); \(400 < 549\), so acute)
- Classification: Acute Scalene Triangle

Triangle 13 (Right Triangle)

- By angle: Right (has 90° angle)
- By sides: Scalene (legs appear different, so no equal sides)
- Classification: Right Scalene Triangle

Triangle 14 (5ft, 5ft, 97°)

- By sides: Isosceles (two sides 5ft)
- By angles: Obtuse (97° > 90°)
- Classification: Obtuse Isosceles Triangle

Triangle 15 (\(\triangle UQR\))

From diagram: \(U\) is midpoint, so \(RU = UT = 8\). \(QR = 8\), \(UQ\): angle at \(Q\) is 90°? Wait, \(\triangle UQR\): angle at \(Q\) is 90°? No, \(\triangle UQR\): angles: 30°, 60°, 90°? Wait, \( \angle URQ = 30^\circ\), \( \angle UQR = 90^\circ\), so right triangle. Sides: \(RU = 8\) (hypotenuse), so \(UQ = 4\), \(QR = 4\sqrt{3}\). So right scalene? Wait, no, \(UQ eq QR\), so right scalene.

Triangle 16 (\(\triangle RST\))

Sides: \(RS = 8\), \(ST = 16\), \(RT = 16\)? Wait, \(RS = 8\), \(ST = 16\), \(RT = 16\)? No, \(RT = 16\), \(RS = 8\), \(ST = 16\)? Wait, \( \triangle RST\): sides \(RS = 8\), \(ST = 16\), \(RT = 16\)? So two sides equal (ST = RT = 16), so isosceles. Angles: angle at \(S\) and \(R\)? Wait, \(RS = 8\), \(ST = RT = 16\). So isosceles with \(ST = RT\), angle at \(S\) and \(R\)? Wait, no, \(ST = RT\), so angles at \(S\) and \(R\) equal? Wait, \(RS = 8\), \(ST = 16\), \(RT = 16\). So isosceles (ST = RT), angle at \(S\) and \(R\) equal. Check angle: \(RS^2 + ST^2 = 64 + 256 = 320\), \(RT^2 = 256\). \(320 eq 256\), so no right angle. \(RT^2 < RS^2 + ST^2\) (256 < 320), so acute. So acute isosceles.

Triangle 17 (\(\triangle SRQ\))

Sides: \(SR = 8\), \(RQ = 8\), \(SQ = 8\)? Wait, \( \angle SRQ = 60^\circ\), \(SR = RQ = 8\), so equilateral? Wait, \(SR = RQ = 8\), angle 60°, so equilateral (all sides equal, all angles 60°). So equilateral (which is also isosceles and acute).

Triangle 18 (\(\triangle PRT\))

Sides: \(PR = 16 + 8 = 24\)? No, \(PT = 16\), \(RT = 16\), \(PR\): wait, \(PT = 16\), \(RT = 16\), so \(PT = RT\), isosceles. Angle at \(T\): 60°? Wait, \(PT = RT = 16\), \( \angle PTR = 60^\circ\), so equilateral (all sides equal, all angles 60°). So equilateral.

Triangle 19 (\(\triangle TQU\))

Sides: \(TQ = 8\), \(QU = 8\), \(TU = 8\)? Wait, \( \angle TQU = 60^\circ\), \(TQ = QU = 8\), so equilateral? Wait, \(TQ = QU = 8\), angle 60°, so equilateral.

But since the user's question is not clear which triangle to classify, assuming triangle 12 (15in, 18in, 20in):

Answer:

For Triangle 12 (15in, 18in, 20in): Acute Scalene Triangle
For Triangle 13 (Right Triangle): Right Scalene Triangle
For Triangle 14 (5ft, 5ft, 97°): Obtuse Isosceles Triangle
For Triangle 15 (\(\triangle UQR\)): Right Scalene Triangle
For Triangle 16 (\(\triangle RST\)): Acute Isosceles Triangle
For Triangle 17 (\(\triangle SRQ\)): Equilateral (Acute Isosceles) Triangle
For Triangle 18 (\(\triangle PRT\)): Equilateral Triangle
For Triangle 19 (\(\triangle TQU\)): Equilateral Triangle

(Assuming the first triangle 12: Acute Scalene Triangle)