__Exam notes for NCERT Class 9 Maths Chapter 7: Triangles – Important Concepts and Formulas CBSE/ SEBA__

__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