Q. In triangle PQR, if angle P = 30 degrees and angle Q = 60 degrees, what is the length of side PR if PQ = 10 cm?
A.
5 cm
B.
8.66 cm
C.
10 cm
D.
12 cm
Show solution
Solution
Using the sine rule: PR/sin(30) = PQ/sin(60) => PR = 10 * sin(30)/sin(60) = 10 * 0.5/√3/2 = 10 * 0.5 * 2/√3 = 10/√3 ≈ 8.66 cm.
Correct Answer:
B
— 8.66 cm
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Q. In triangle PQR, if angle P = 30 degrees and angle Q = 90 degrees, what is angle R?
A.
60 degrees
B.
30 degrees
C.
90 degrees
D.
120 degrees
Show solution
Solution
Angle R = 180 - (30 + 90) = 60 degrees.
Correct Answer:
A
— 60 degrees
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Q. In triangle PQR, if angle P = 30 degrees and angle Q = 90 degrees, what is the length of side PR if PQ = 10 cm?
A.
5 cm
B.
10 cm
C.
15 cm
D.
20 cm
Show solution
Solution
In a right triangle, the side opposite the 30-degree angle is half the hypotenuse. Therefore, PR = PQ = 10 cm.
Correct Answer:
B
— 10 cm
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Q. In triangle PQR, if angle P = 45 degrees and angle Q = 45 degrees, what is the type of triangle?
A.
Scalene
B.
Isosceles
C.
Equilateral
D.
Right
Show solution
Solution
Since two angles are equal, triangle PQR is an isosceles triangle.
Correct Answer:
B
— Isosceles
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Q. In triangle PQR, if angle P = 90 degrees and PQ = 6 cm, PR = 8 cm, what is the length of QR?
A.
10 cm
B.
12 cm
C.
14 cm
D.
16 cm
Show solution
Solution
Using the Pythagorean theorem, QR = √(PQ² + PR²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Correct Answer:
A
— 10 cm
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Q. In triangle PQR, if PQ = 12 cm, PR = 16 cm, and QR = 20 cm, is triangle PQR similar to triangle ABC with sides 3 cm, 4 cm, and 5 cm?
A.
Yes
B.
No
C.
Only if angles are equal
D.
Not enough information
Show solution
Solution
The sides of triangle PQR are in the ratio 12:16:20 = 3:4:5, which matches triangle ABC, confirming similarity.
Correct Answer:
A
— Yes
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Q. In triangle PQR, if PQ = 12 cm, PR = 9 cm, and QR = 15 cm, which criterion can be used to prove that triangle PQR is not congruent to triangle STU with sides ST = 12 cm, SU = 9 cm, and TU = 14 cm?
A.
SSS
B.
SAS
C.
ASA
D.
AAS
Show solution
Solution
Triangle PQR cannot be congruent to triangle STU by SSS since the third side QR (15 cm) is not equal to TU (14 cm).
Correct Answer:
A
— SSS
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Q. In triangle PQR, if PQ = 5 cm, QR = 12 cm, and PR = 13 cm, is triangle PQR a right triangle?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if angle P is 90 degrees
Show solution
Solution
Using the Pythagorean theorem, 5^2 + 12^2 = 25 + 144 = 169 = 13^2, so triangle PQR is a right triangle.
Correct Answer:
A
— Yes
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Q. In triangle PQR, if PQ = 5 cm, QR = 12 cm, and PR = 13 cm, what type of triangle is it?
A.
Acute
B.
Obtuse
C.
Right
D.
Equilateral
Show solution
Solution
Using the Pythagorean theorem, 5^2 + 12^2 = 25 + 144 = 169, which equals 13^2. Therefore, triangle PQR is a right triangle.
Correct Answer:
C
— Right
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Q. In triangle PQR, if PQ = 6 cm, PR = 8 cm, and QR = 10 cm, is triangle PQR a right triangle?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if angle P is 90°
Show solution
Solution
Using the Pythagorean theorem, if QR² = PQ² + PR², then 10² = 6² + 8², which gives 100 = 36 + 64 = 100. Therefore, triangle PQR is a right triangle.
Correct Answer:
A
— Yes
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Q. In triangle PQR, if PQ = 6 cm, QR = 8 cm, and PR = 10 cm, what type of triangle is it?
A.
Acute
B.
Obtuse
C.
Right
D.
Equilateral
Show solution
Solution
It is a right triangle because 6^2 + 8^2 = 36 + 64 = 100 = 10^2.
Correct Answer:
C
— Right
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Q. In triangle PQR, if PQ = 7 cm, PR = 24 cm, and QR = 25 cm, what type of triangle is it?
A.
Acute
B.
Obtuse
C.
Right
D.
Equilateral
Show solution
Solution
Using the Pythagorean theorem, 7² + 24² = 49 + 576 = 625 = 25², thus triangle PQR is a right triangle.
Correct Answer:
C
— Right
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Q. In triangle PQR, if PQ = 8 cm, PR = 6 cm, and angle P = 90 degrees, what is the length of QR?
A.
10 cm
B.
12 cm
C.
14 cm
D.
16 cm
Show solution
Solution
Using the Pythagorean theorem, QR = √(PQ² + PR²) = √(8² + 6²) = √(64 + 36) = √100 = 10 cm.
Correct Answer:
A
— 10 cm
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Q. In triangle PQR, if PQ = 8 cm, PR = 6 cm, and QR = 10 cm, is triangle PQR a right triangle?
A.
Yes
B.
No
C.
Cannot be determined
D.
Only if angle P is 90 degrees
Show solution
Solution
Using the Pythagorean theorem, if PQ² + PR² = QR², then it is a right triangle. 8² + 6² = 64 + 36 = 100 = 10².
Correct Answer:
A
— Yes
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Q. In triangle PQR, if PQ = 8 cm, QR = 6 cm, and PR = 10 cm, what type of triangle is it?
A.
Equilateral
B.
Isosceles
C.
Scalene
D.
Right
Show solution
Solution
Since all sides are of different lengths, triangle PQR is a scalene triangle.
Correct Answer:
C
— Scalene
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Q. In triangle RST, if RS = 10 cm, ST = 24 cm, and RT = 26 cm, what is the perimeter of triangle RST?
A.
50 cm
B.
60 cm
C.
70 cm
D.
80 cm
Show solution
Solution
The perimeter is the sum of all sides: 10 + 24 + 26 = 60 cm.
Correct Answer:
A
— 50 cm
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Q. In triangle RST, if RS = 5 cm, ST = 12 cm, and RT = 13 cm, is triangle RST a right triangle?
A.
Yes
B.
No
C.
Only if angle R is 90 degrees
D.
Only if angle S is 90 degrees
Show solution
Solution
Using the Pythagorean theorem, check if 5^2 + 12^2 = 13^2. 25 + 144 = 169, which is true. Therefore, triangle RST is a right triangle.
Correct Answer:
A
— Yes
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Q. In triangle STU, if angle S = 30 degrees and angle T = 45 degrees, what is the measure of angle U?
A.
45 degrees
B.
30 degrees
C.
105 degrees
D.
75 degrees
Show solution
Solution
The sum of angles in a triangle is 180 degrees. Therefore, angle U = 180 - (30 + 45) = 105 degrees.
Correct Answer:
D
— 75 degrees
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Q. In triangle STU, if angle S = 30 degrees and angle T = 60 degrees, what is the measure of angle U?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
Show solution
Solution
The sum of angles in a triangle is 180 degrees. Therefore, angle U = 180 - (30 + 60) = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. In triangle STU, if angle S = 30 degrees and angle T = 70 degrees, what is the length of side SU if ST = 10 cm and TU = 15 cm?
A.
7.5 cm
B.
10 cm
C.
12.5 cm
D.
15 cm
Show solution
Solution
Using the Law of Sines: SU / sin(80) = 10 / sin(30). Therefore, SU = 10 * sin(80) / sin(30) = 10 * 0.9848 / 0.5 = 19.696 cm.
Correct Answer:
C
— 12.5 cm
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Q. In triangle STU, if angle S = 45 degrees and angle T = 45 degrees, what is the type of triangle STU?
A.
Acute
B.
Obtuse
C.
Right
D.
Isosceles
Show solution
Solution
Since two angles are equal, triangle STU is isosceles.
Correct Answer:
D
— Isosceles
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Q. In triangle STU, if ST = 12 cm, TU = 16 cm, and SU = 20 cm, what is the perimeter of triangle STU?
A.
28 cm
B.
36 cm
C.
40 cm
D.
48 cm
Show solution
Solution
The perimeter of triangle STU is ST + TU + SU = 12 + 16 + 20 = 48 cm.
Correct Answer:
B
— 36 cm
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Q. In triangle STU, if ST = 7 cm, TU = 24 cm, and SU = 25 cm, is triangle STU a right triangle?
A.
Yes
B.
No
C.
Only if angle S is 90 degrees
D.
Only if angle T is 90 degrees
Show solution
Solution
Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625, which equals 25^2. Therefore, triangle STU is a right triangle.
Correct Answer:
A
— Yes
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Q. In triangle UVW, if angle U = 45 degrees and angle V = 45 degrees, what type of triangle is it?
A.
Equilateral
B.
Isosceles
C.
Scalene
D.
Right
Show solution
Solution
Since two angles are equal (45 degrees each), triangle UVW is an isosceles triangle.
Correct Answer:
B
— Isosceles
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Q. In triangle XYZ, if angle X = 45 degrees and angle Y = 45 degrees, what type of triangle is it?
A.
Scalene
B.
Isosceles
C.
Equilateral
D.
Right
Show solution
Solution
Since two angles are equal, triangle XYZ is an isosceles triangle.
Correct Answer:
B
— Isosceles
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Q. In triangle XYZ, if angle X = 90 degrees, angle Y = 45 degrees, and side XY = 10 units, what is the length of side XZ?
A.
10√2 units
B.
5√2 units
C.
10 units
D.
5 units
Show solution
Solution
In a 45-45-90 triangle, the sides opposite the 45-degree angles are equal, and the hypotenuse is √2 times the length of each leg. Thus, XZ = XY√2 = 10√2 units.
Correct Answer:
A
— 10√2 units
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Q. In triangle XYZ, if angle X is 30 degrees and angle Y is 60 degrees, what is angle Z?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
Show solution
Solution
The sum of the angles in a triangle is 180 degrees. Therefore, angle Z = 180 - (30 + 60) = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. In triangle XYZ, if XY = 5 cm, YZ = 12 cm, and XZ = 13 cm, is triangle XYZ a right triangle?
A.
Yes
B.
No
C.
Not enough information
D.
Only if XY is the longest side
Show solution
Solution
Triangle XYZ is a right triangle because 5² + 12² = 25 + 144 = 169 = 13².
Correct Answer:
A
— Yes
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Q. In triangle XYZ, if XY = 7 cm, YZ = 24 cm, and XZ = 25 cm, is triangle XYZ a right triangle?
A.
Yes
B.
No
C.
Only if angle Y is 90 degrees
D.
Only if angle Z is 90 degrees
Show solution
Solution
Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625 = 25^2, thus triangle XYZ is a right triangle.
Correct Answer:
A
— Yes
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Q. In triangle XYZ, if XY = 8 cm, XZ = 6 cm, and YZ = 10 cm, what type of triangle is it?
A.
Equilateral
B.
Isosceles
C.
Scalene
D.
Right
Show solution
Solution
Triangle XYZ is scalene because all sides have different lengths.
Correct Answer:
C
— Scalene
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