Exam notes for NCERT Class 9 Maths Chapter 7: Triangles – Important Concepts and Formulas CBSE/ SEBA
Here are the important concepts and formulas for Class 9 Maths Chapter 7: Triangles.
- Angle Sum Property of a Triangle:
The sum of the interior angles of a triangle is always 180°.
- Exterior Angle Property of a Triangle:
The measure of an exterior angle of a triangle is the sum of the measures of the two non-adjacent interior angles.
- Congruence of Triangles:
Two triangles are said to be congruent if their corresponding sides and angles are equal.
- Congruence Conditions:
SAS:Side-Angle-Side, AAS:Angle-Angle-Side, SSS:Side-Side-Side, RHS:Right-Angle-Hypotenuse-Side
- Pythagoras Theorem:
In a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
- The similarity of triangles:
Two triangles are said to be similar if their corresponding angles are equal, and their corresponding sides are in the same ratio.
- Basic Proportionality Theorem:
If a line parallel to one side of a triangle intersects the other two sides in distinct points, then it divides the two sides proportionally.
- Perimeter:
The perimeter of a polygon is the sum of the lengths of its sides.
- Heron’s Formula:
Heron’s formula is used to find the area of a triangle when we know the lengths of its three sides.
𝑠 = (𝑎 + 𝑏 + 𝑐)/2
Area of triangle ∆ =√s(s – a)(s – b)(s -c)
- Median:
A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side.
- Altitude:
An altitude of a triangle is a line segment through a vertex and perpendicular to the opposing side.
- Area of a triangle using base and height:
Area of a triangle ∆ = (1/2) × Base × Height
- Pythagoras’ Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- The Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of its interior opposite angles.
- Similar Triangles: Two triangles are said to be similar when the corresponding angles are equal and the corresponding sides are proportional.
- The Angle Bisector Theorem: In a triangle, the angle bisector of a given angle divides the opposite side into two segments that are proportional to the other two sides of the triangle.
- The Median of a Triangle: A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In any triangle, the three medians intersect at a point called the centroid.
- Area of a Triangle: The area of a triangle can be calculated by using the formula A = 1/2 b x h, where b is the triangle’s base, and h is the triangle’s height.
- Congruence of Triangles: Two triangles are said to be congruent when all their corresponding sides and angles are equal.
- Euler’s Formula: In a triangle, the sum of the distances from any point inside the triangle to its three sides is equal to the perimeter of the triangle.
- Trigonometric Ratios: In a right-angled triangle, the ratio of the lengths of any two sides gives trigonometric ratios such as sine, cosine, and tangent.
Exam MCQs with answers for NCERT Class 9 Maths Chapter 7: Triangles – Important Concepts and Formulas CBSE/ SEBA
Which of the following is not the property of an isosceles triangle?
- A) Two sides are equal
- B) Two angles are equal
- C) The sum of all angles is 180 degrees
- D) The perpendicular bisector of the base intersects the vertex angle bisector
Answer: C
In a triangle ABC, altitude BD is drawn on AC. If AC = 10 cm, BD = 6 cm, find BC.
- A) 8 cm
- B) 12 cm
- C) 14 cm
- D) 16 cm
Answer: B
In a right-angled triangle ABC, ∠B = 90 degrees and AB = 5 cm. If BC: AC = 3:4, then find BC.
- A) 3 cm
- B) 4 cm
- C) 7.5 cm
- D) 10 cm
Answer: C
In a triangle ABC, if ∠A = 60 degrees and BC = 4 cm, then what is the measure of the altitude drawn on BC?
- A) 2 cm
- B) 2√3 cm
- C) 4 cm
- D) 4√3 cm
Answer: B
In a triangle ABC, if AB = AC and the measure of ∠BAC is 80 degrees, then what is the measure of ∠BCA?
- A) 40 degrees
- B) 50 degrees
- C) 70 degrees
- D) 80 degrees
Answer: A
In a triangle ABC, AD is the median from A to BC. If AB = 6 cm and AC = 8 cm, then what is the length of AD?
- A) 4 cm
- B) 4.5 cm
- C) 5 cm
- D) 5.5 cm
Answer: C
In a triangle ABC, if the internal bisector of ∠A meets BC at D, then what is the ratio of BD: to DC?
- A) AB:AC
- B) AC:AB
- C) BC:AC
- D) AC:BC
Answer: B
In ΔABC, ∠ABC = 90°, AB = 5 cm and BC = 12 cm. Find AC.
- A) 11 cm
- B) 13 cm
- C) 17 cm
- D) 15 cm
Answer: B) 13 cm
The area of a triangular field is 48 square metres. If the base of the triangle is 8 metres, find the height.
- A) 6 metres
- B) 12 metres
- C) 24 metres
- D) 3 metres
Answer: A) 6 metres
If two sides of a triangle are 6 cm and 8 cm, then the third side can be:
- A) 2 cm
- B) 12 cm
- C) 9 cm
- D) 14 cm
Answer: D) 14 cm
The sum of the lengths of any two sides of a triangle is always:
- A) Equal to the third side
- B) Greater than the third side
- C) Less than the third side
- D) None of the above
Answer: B) Greater than the third side
In a right-angled triangle, the side opposite to the right angle is called:
- A) Hypotenuse
- B) Altitude
- C) Base
- D) None of the above
Answer: A) Hypotenuse
What is the sum of the angles of a triangle?
- a) 90 degrees
- b) 180 degrees
- c) 270 degrees
- d) 360 degrees
Answer: b) 180 degrees
In a right-angle triangle, if one acute angle is x, what is the other acute angle?
- a) 90-x
- b) x/2
- c) 180-x
- d) x+90
Answer: a) 90-x
If the angle bisector of a triangle divides the opposite side in the ratio m:n, then the length of the side adjacent to the smaller part is proportional to:
- a) n
- b) m+n
- c) m-n
- d) m
Answer: d) m
If in two triangles, the corresponding angles are equal, then their corresponding sides are proportional. This theorem is known as:
- a) Angle-angle-angle theorem
- b) Side-angle-side theorem
- c) Angle-side-angle theorem
- d) Side-side-side theorem
Answer: a) Angle-angle-angle theorem
In a triangle ABC, if AB=AC, what is the measure of angle BAC?
- a) 60 degrees
- b) 30 degrees
- c) 90 degrees
- d) 45 degrees
Answer: d) 45 degrees
If in a triangle, one angle is two-thirds of the sum of the other two angles, then the largest angle of the triangle is:
- a) 60 degrees
- b) 90 degrees
- c) 120 degrees
- d) 150 degrees
Answer: c) 120 degrees
If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. This theorem is called:
- a) Sine rule
- b) Cosine rule
- c) Isosceles triangle theorem
- d) Triangle inequality theorem
Answer: d) Triangle inequality theorem
If a line is drawn parallel to one side of a triangle, then the other two sides are divided in the same ratio. This theorem is called:
- a) Basic proportionality theorem
- b) Angle bisector theorem
- c) Mid-point theorem
- d) Pythagoras’ theorem
Answer: a) Basic proportionality theorem
Which of the following is not possible?
- An obtuse triangle with all its angles equal
- A right-angled isosceles triangle
- An equilateral triangle with all its sides unequal
- A scalene triangle with two sides equal
Answer: c
In the given figure, AD and BC are equal sides of triangles ABC and AED, respectively. The value of x is:
- 45 degrees
- 60 degrees
- 75 degrees
- 90 degrees
Answer: b
In a right triangle ABC, where AB = BC, point D is on AC such that BD is perpendicular to AC. If AD = 3 and CD = 9, then what is AB?
- 12
- 15
- 18
- 21
Answer: b
Two similar triangles have areas in the ratio 4:9. If the perimeter of the smaller triangle is 10 cm, then what is the perimeter of the larger triangle?
- 12.5 cm
- 15 cm
- 22.5 cm
- 25 cm
Answer: d
In the given figure, if AB/AD = 3/2 and BD/DC = 5/3, then what is the value of BC/DE?
- 15/8
- 8/15
- 5/6
- 6/5
Answer: b
In an isosceles triangle ABC, AB = AC and D is a midpoint of BC. If angle ADB = 60 degrees, then what is the measure of angle BAC?
- 60 degrees
- 75 degrees
- 90 degrees
- 120 degrees
Answer: d
In the given figure, ABCD is a square and BCE is an equilateral triangle. If AB = 6 cm, then what is the length of EC?
- 6 cm
- 6√2 cm
- 9 cm
- 9√2 cm
Answer: b
In a triangle ABC, if AB=3cm, BC=4cm and AC=5cm, what is the length of the median AD from vertex A to side BC?
Answer: 2.5cm
In a triangle ABC, if AB=3cm, BC=4cm and AC=5cm, what is the length of the altitude from vertex A to side BC?
Answer: 2.4cm
The sum of the lengths of any two sides of a triangle is always _____ than the length of the third side.
Answer: greater
In a triangle ABC, the sum of the angles at vertices A and B is 120 degrees. What is the measure of the angle at vertex C?
Answer: 60 degrees
In a right-angled triangle ABC, the length of the hypotenuse is 10cm and the length of one of the legs is 6cm. What is the length of the other leg?
Answer: 8cm
In triangle PQR, if PQ=10cm, QR=12cm and RP=14cm, what is the area of the triangle?
Answer: 58.78 sq cm
If two triangles are congruent, then their corresponding angles and ____ are also congruent.
Answer: sides
In triangle XYZ, if the angle at vertex X is 50 degrees and the side opposite to it is 8cm long, what is the length of the median from vertex X to side YZ?
Answer: 6.93cm
In a triangle ABC, if AB=10cm, BC=12cm and the angle at vertex C are 60 degrees, what is the length of the altitude from vertex C to side AB?
Answer: 6cm
In triangle XYZ, if the angle at vertex Z is 90 degrees and the length of the altitude from vertex Z to side XY is 7cm, what is the area of the triangle?
Answer: 24.5 sq cm
The sum of the angles of a triangle is ____.
Answer: 180 degrees
A triangle is said to be acute if all its angles are less than ____.
Answer: 90 degrees
A triangle is said to be obtuse if one of its angles is greater than ____.
Answer: 90 degrees
A triangle is said to be equilateral if all its sides are ____.
Answer: Equal
The sum of two sides of a triangle is always ____ than the third side.
Answer: greater
The angle opposite to the longest side of a triangle is ____.
Answer: largest
In an isosceles triangle, the angles opposite to the equal sides are ____.
Answer: Equal
The perpendicular line from the vertex of a right-angled triangle to the hypotenuse is called the ____.
Answer: altitude
The median of a triangle joins the vertex to the ____ of the opposite side.
Answer: midpoint
The area of a triangle is ____ of the product of its base and height.
Answer: half
