🔺 What is Triangle?
English: A triangle is a closed shape that has 3 sides and 3 angles.
Roman Urdu: Triangle ek band shape hoti hai jis mein 3 sides aur 3 angles hote hain.
👉 Example: Agar ek shape ke 3 straight lines ko mila dein, toh woh triangle ban jata hai.
✅ Types of Triangles (Based on Sides):
- Equilateral Triangle – All 3 sides are equal (Roman Urdu: Teeno sides barabar hoti hain).
- Isosceles Triangle – Two sides equal (Roman Urdu: Do sides barabar hoti hain).
- Scalene Triangle – No sides equal (Roman Urdu: Koi bhi side barabar nahi hoti).
✅ Types of Triangles (Based on Angles):
- Acute Triangle – All angles < 90°
- Right Triangle – One angle = 90°
- Obtuse Triangle – One angle > 90°
📐 What is Trigonometry?
English: Trigonometry is a branch of mathematics that studies the relationship between the angles and sides of a triangle.
Roman Urdu: Trigonometry math ka woh hissa hai jo triangle ke sides aur angles ke darmiyan relationship ko samjhata hai.
👉 Mostly Right-Angle Triangle (jismein ek angle 90° hota hai) mein use hoti hai.
🔑 Difference between Triangle and Trigonometry
- Triangle = Shape (geometry ka part)
- Trigonometry = Study of relationships between triangle’s angles and sides (mathematics ka part)
Roman Urdu:
- Triangle sirf ek shape hai jisme 3 sides aur 3 angles hote hain.
- Trigonometry ek subject hai jo triangle ke sides aur angles ka relation batata hai.
📊 Trigonometric Ratios / Functions
In a Right Triangle, we define special ratios:
- Sine (sin θ) = Opposite Side / Hypotenuse
- Cosine (cos θ) = Adjacent Side / Hypotenuse
- Tangent (tan θ) = Opposite Side / Adjacent Side
- Cosecant (csc θ) = 1/sin θ = Hypotenuse / Opposite Side
- Secant (sec θ) = 1/cos θ = Hypotenuse / Adjacent Side
- Cotangent (cot θ) = 1/tan θ = Adjacent Side / Opposite Side
Roman Urdu:
- Opposite side = angle ke saamne wali side
- Adjacent side = angle ke paas wali side
- Hypotenuse = sabse lambi side (90° ke opposite)
⭐ Main Points Summary:
- Triangle → 3 sides, 3 angles
- Types → Based on sides (equilateral, isosceles, scalene), based on angles (acute, right, obtuse)
- Trigonometry → Math ka branch jo triangle ke sides aur angles ko relate karta hai
- Trigonometric Ratios → sin, cos, tan, csc, sec, cot
:
🔹 Triangle & Trigonometry PRACTICE MCQs with Answers and Explanations
1) The sum of the angles in a triangle is always:
A) 90°
B) 270°
C) 180° ✅
D) 360°
Explanation: Har triangle mein 3 angles hote hain aur unka sum hamesha 180° hota hai (Triangle Angle Sum Property).
2) A triangle with all three sides equal is called:
A) Scalene triangle
B) Isosceles triangle
C) Equilateral triangle ✅
D) Right triangle
Explanation: Equilateral triangle wo hoti hai jisme 3 sides barabar aur har angle 60° hota hai.
3) A triangle with two equal sides is called:
A) Isosceles triangle ✅
B) Scalene triangle
C) Right triangle
D) Equilateral triangle
Explanation: Isosceles triangle wo hoti hai jisme 2 sides barabar aur unke opposite angles bhi barabar hote hain.
4) In a right triangle, the side opposite the right angle is called the:
A) Adjacent side
B) Opposite side
C) Hypotenuse ✅
D) Base
Explanation: Right triangle mein jo side 90° ke saamne hoti hai, usay hypotenuse kehte hain. Ye hamesha sabse badi side hoti hai.
5) Which of the following is NOT a trigonometric ratio?
A) Sine
B) Cosine
C) Tangent
D) Gradient ✅
Explanation: Trigonometric ratios ke names hain: sin, cos, tan, cosec, sec, cot. “Gradient” trigonometry ka ratio nahi hai, ye slope ke liye use hota hai.
6) If sin θ = 1, then θ equals:
A) 30°
B) 45°
C) 60°
D) 90° ✅
Explanation: sin θ = Opposite/Hypotenuse. Sin ka maximum value 1 hota hai aur ye tab hota hai jab θ = 90°.
7) If a triangle has angles 30°, 60°, 90°, then the sides are in the ratio:
A) 1 : 2 : √3
B) 1 : √3 : 2 ✅
C) 2 : √3 : 1
D) √2 : 1 : 2
Explanation: Ek 30°–60°–90° triangle ki property hoti hai ke uski sides ratio mein hoti hain:
Opposite 30° = 1, Opposite 60° = √3, Hypotenuse = 2.
8) cos 0° equals:
A) 1/2
B) 0
C) 1 ✅
D) √2/2
Explanation: Trigonometric values yaad rakhne wali table ke mutabiq:
cos 0° = 1, sin 0° = 0, tan 0° = 0.
9) tan 45° equals:
A) 1/2
B) 0
C) √3
D) 1 ✅
Explanation: tan θ = Opposite / Adjacent. 45° angle par dono sides equal hoti hain → tan 45° = 1.
10) If two angles of a triangle are 40° and 80°, the third angle is:
A) 50°
B) 60° ✅
C) 40°
D) 70°
Explanation: Triangle angle sum = 180°.
Third angle = 180° − (40° + 80°) = 180° − 120° = 60°.
1) The sum of the interior angles of any triangle is:
A) 90°
B) 120°
C) 180° ✅
D) 270°
Answer: C) 180°
Explanation: According to the Triangle Angle-Sum Theorem, the sum of all interior angles of a triangle is always 180°.
2) A triangle with one angle equal to 90° is called:
A) Acute Triangle
B) Obtuse Triangle
C) Right Triangle ✅
D) Equilateral Triangle
Answer: C) Right Triangle
Explanation: A Right Triangle has exactly one right angle (90°) and the largest side opposite to it is called the hypotenuse.
3) If two angles of a triangle are 47° and 63°, the third angle is:
A) 70° ✅
B) 60°
C) 80°
D) 90°
Answer: A) 70°
Explanation: By the Triangle Angle-Sum Property,
Third angle = 180° − (47° + 63°) = 70°.
