Triangle MCQ Quiz - Objective Question with Answer for Triangle - Download Free PDF
Last updated on Jul 11, 2025
Latest Triangle MCQ Objective Questions
Triangle Question 1:
The sides (in cm) of a right triangle are (x − 13), (x − 26) and x. Its area (in cm2) is:
Answer (Detailed Solution Below)
Triangle Question 1 Detailed Solution
Given:
Sides of right triangle: (x − 13), (x − 26), and x cm
Formula used:
Pythagoras Theorem: Hypotenuse2 = Base2 + Height2
Area = (1/2) × Base × Height
Calculation:
Hypotenuse = x, Base = (x - 26), Height = (x - 13)
Assume hypotenuse = x
⇒ x2 = (x − 13)2 + (x − 26)2
⇒ x2 = (x2 − 26x + 169) + (x2 − 52x + 676)
⇒ x2 = 2x2 − 78x + 845
⇒ Bring all terms to one side:
⇒ x2 − 2x2 + 78x − 845 = 0
⇒ −x2 + 78x − 845 = 0
⇒ x2 − 78x + 845 = 0
Solve using quadratic formula:
x = [78 ± √(782 − 4 × 1 × 845)] ÷ 2
x = [78 ± √(6084 − 3380)] ÷ 2
x = [78 ± √2704] ÷ 2
x = [78 ± 52] ÷ 2
⇒ x = (78 + 52)/2 = 130 ÷ 2 = 65 (valid)
⇒ x = (78 − 52)/2 = 26 ÷ 2 = 13 (invalid as sides become 0)
If x = 13, then one side is (x - 13) = (13 - 13) = 0, which is not possible for a triangle. So, x ≠ 13.
So, x = 65
Now, find the lengths of the sides:
Side 1 = x - 13 = 65 - 13 = 52 cm
Side 2 = x - 26 = 65 - 26 = 39 cm
Hypotenuse = x = 65 cm
Area = (1/2) × 39 × 52 = 1014 cm2
∴ Area = 1014 cm2
Triangle Question 2:
Comprehension:
ABC is a triangle right-angled at B. The perimeter of the triangle is 24 cm and the difference between the sum of the perpendicular sides and the hypotenuse is 4 cm.
A circle is inscribed in the triangle. What is its radius?
Answer (Detailed Solution Below)
Triangle Question 2 Detailed Solution
Given:
p + b + h = 24
p + b - h = 4
Formula used:
The radius (r) of the inscribed circle in a right-angled triangle is given by the formula:
r =
Where: a and b are the perpendicular sides of the triangle
c is the hypotenuse of the triangle
Calculation:
p + b + h = 24 cm
p + b - h = 4 cm
The radius (r) of the inscribed circle in a right-angled triangle is given by the formula:
r =
r = 4/2 = 2 cm
Therefore, the radius of the inscribed circle is: 2 cm.
Triangle Question 3:
Comprehension:
ABC is a triangle right-angled at B. The perimeter of the triangle is 24 cm and the difference between the sum of the perpendicular sides and the hypotenuse is 4 cm.
What is the area of the triangle ABC?
Answer (Detailed Solution Below)
Triangle Question 3 Detailed Solution
Given:
p + b + h = 24
p + b - h = 4
Formula used:
The radius (r) of the inscribed circle in a right-angled triangle is given by the formula:
r =
Where: p and b are the perpendicular sides of the triangle
Area of triangle = radius (r) of the inscribed circle × semi - perimeter.
c is the hypotenuse of the triangle
Calculation:
p + b + h = 24 cm
p + b - h = 4 cm
The radius (r) of the inscribed circle in a right-angled triangle is given by the formula:
r =
r = 4/2 = 2 cm
Area of triangle = 2 × 24/2 = 24 cm2
∴ The correct anser is option 2.
Triangle Question 4:
In a triangle ABC, AB + BC = 7.1 cm, BC + CA = 12.1 cm and CA + AB = 7.2 cm. What is the area of the triangle?
Answer (Detailed Solution Below)
Triangle Question 4 Detailed Solution
Given:
AB + BC = 7.1 cm
BC + CA = 12.1 cm
CA + AB = 7.2 cm
Calculation:
Let the sides of the triangle be:
AB = a
BC = b
CA = c
We are given the following system of equations:
a + b = 7.1
b + c = 12.1
c + a = 7.2
(a + b) + (b + c) + (c + a) = 7.1 + 12.1 + 7.2
2a + 2b + 2c = 26.4
a + b + c = 13.2
c = 13.2 - 7.1 = 6.1
a = 13.2 - 12.1 = 1.1
b = 13.2 - 7.2 = 6.0
Heron’s formula for the area of a triangle is given by:
Where s is the semi-perimeter of the triangle, given by:
s =
Therefore, the area of the triangle is: 3.3 cm²
Triangle Question 5:
The number of triangles in the figure is
Answer (Detailed Solution Below)
Triangle Question 5 Detailed Solution
Total number of triangle is:
Hence, the correct answer is "Option 2'.
Top Triangle MCQ Objective Questions
If the side of an equilateral triangle is increased by 34%, then by what percentage will its area increase?
Answer (Detailed Solution Below)
Triangle Question 6 Detailed Solution
Download Solution PDFGiven:
The sides of an equilateral triangle are increased by 34%.
Formula used:
Effective increment % = Inc.% + Inc.% + (Inc.2/100)
Calculation:
Effective increment = 34 + 34 + {(34 × 34)/100}
⇒ 68 + 11.56 = 79.56%
∴ The correct answer is 79.56%.
In an isosceles triangle ABC, if AB = AC = 26 cm and BC = 20 cm, find the area of triangle ABC.
Answer (Detailed Solution Below)
Triangle Question 7 Detailed Solution
Download Solution PDFGiven:
In an isosceles triangle ABC,
AB = AC = 26 cm and BC = 20 cm.
Calculations:
In this triangle ABC,
∆ADC = 90° ( Angle formed by a line from opposite vertex to unequal side at mid point in isosceles triangle is 90°)
So,
AD² + BD² = AB² (by pythagoras theorem)
⇒ AD² = 576
⇒ AD = 24
Area of triangle = ½(base × height)
⇒ ½(20 × 24) (Area of triangle = (1/2) base × height)
⇒ 240 cm²
∴ The correct choice is option 2.
If the perimeter of a triangle is 28 cm and its inradius is 3.5 cm, what is its area?
Answer (Detailed Solution Below)
Triangle Question 8 Detailed Solution
Download Solution PDFSemi-perimeter of the triangle (s) = 28/2 = 14
As we know,
Area of triangle = Inradius × S = 3.5 × 14 = 49 cm2
An equilateral triangle ABC is inscribed in a circle with centre O. D is a point on the minor arc BC and ∠CBD = 40º. Find the measure of ∠BCD.
Answer (Detailed Solution Below)
Triangle Question 9 Detailed Solution
Download Solution PDFGiven:
An equilateral triangle ABC is inscribed in a circle with centre O
∠CBD = 40º
Concept used:
The sum of the opposite angles of a cyclic quadrilateral = 180°
The sum of all three angles of a triangle = 180°
Calculation:
∠ABC = ∠ACB = ∠BAC = 60° [∵ ΔABC is an equilateral triangle]
Also, ∠BAC + ∠BDC = 180°
⇒ 60° + ∠BDC = 180°
⇒ ∠BDC = 180° - 60° = 120°
Also, ∠CBD + ∠BDC + ∠BCD = 180°
⇒ 40° + 120° + ∠BCD = 180°
⇒ ∠BCD = 180° - 40° - 120° = 20°
∴ The value of ∠BCD is 20°
In a ΔABC, points P, Q and R are taken on AB, BC and CA, respectively, such that BQ = PQ and QC = QR. If ∠BAC = 75º, what is the measure of ∠PQR (in degrees)?
Answer (Detailed Solution Below)
Triangle Question 10 Detailed Solution
Download Solution PDFShortcut Trick
∠BAC = 75º
∠ABC + ∠ACB + ∠BAC = 180°
∠ABC + ∠ACB + 75° = 180°
∠ABC + ∠ACB = 180° - 75° = 105°
Let, ∠ABC = ∠PBQ = 70° and ∠ACB = ∠RCQ = 35°
So, ∠PQR = 180° - (∠PQB + ∠RQC)
= 180° - [(180° - 2∠PBQ) + (180° - 2∠RCQ) [∵ BQ = PQ; QC = QR]
= 180° - [(180° - 2 × 70°) + (180° - 2 × 35°)]
= 180° - (40° + 110°)
= 180° - 150°
= 30°
Alternate Method
Given:
In a ΔABC, ∠BAC = 75º
BQ = PQ and QC = QR
Concept used:
The sum of all three angles of a triangle = 180°
The sum of all angles on a straight line = 180°
Calculation:
Let, ∠ABC = x and ∠ACB = y
So, ∠ABC = ∠PBQ = ∠QPB = x [∵ BQ = PQ]
∠ACB = ∠RCQ = ∠QRC = y [QC = QR]
In ΔABC, ∠ABC + ∠ACB + ∠BAC = 180°
⇒ x + y + 75° = 180°
⇒ x + y = 180° - 75° = 105° .....(1)
For ΔBPQ and ΔCRQ,
(∠PBQ + ∠QPB + ∠PQB) + (∠RCQ + ∠QRC + ∠RQC) = 180° + 180° = 360°
⇒ (x + x + ∠PQB) + (y + y + ∠RQC) = 360°
⇒ 2x + 2y + ∠PQB + ∠RQC = 360°
⇒ 2 (x + y) + ∠PQB + ∠RQC = 360°
⇒ (2 × 105°) + ∠PQB + ∠RQC = 360° [∵ x + y = 105°]
⇒ ∠PQB + ∠RQC = 360° - 210° = 150° .....(2)
Also, ∠PQB + ∠RQC + ∠PQR = 180°
⇒ 150° + ∠PQR = 180° [∵ ∠PQB + ∠RQC = 150°]
⇒ ∠PQR = 180° - 150° = 30°
∴ The measure of ∠PQR (in degrees) is 30°
The perimeter and one of two equal sides of an isosceles triangle are 72 cm and 20 cm respectively. Area of the triangle is:
Answer (Detailed Solution Below)
Triangle Question 11 Detailed Solution
Download Solution PDFGiven,
One of two equal sides of an isosceles triangle, a = 20 cm
Perimeter of the triangle = 72 cm
Formula:
Perimeter of an isosceles triangle = 2a + b
Area of an isosceles triangle = (b/4) × √(4a2 – b2)
Calculation:
Let a = 20 cm
2a + b = 72
⇒ 2 × 20 + b = 72
⇒ 40 + b = 72
⇒ b = 72 – 40
⇒ b = 32
Area of isosceles triangle = (32/4) × √(4 × 202 – 322)
⇒ 8 × √(4 × 400 – 1024)
⇒ 8 × √(1600 – 1024)
⇒ 8 × √576
⇒ 8 × 24
⇒ 192 cm2
∴ Area of the triangle is 192 cm2.
Alternate solution
Third side = 72 – 2 × 20 = 72 – 40 = 32
Semi perimeter, s = 72/2 = 36
Now,
Area = √[s (s – a) (s – b) (s – c)] = √[36(36 – 32)(36 - 20)(36 - 20)] = √(36 × 4 × 16 × 16) = 16 × 4 × 3 = 192 cm2
In an equilateral ΔABC, the medians AD, BE and CF intersect to each other at point G. If the area of quadrilateral BDGF is 12√3 cm2, then the side of ΔABC is:
Answer (Detailed Solution Below)
Triangle Question 12 Detailed Solution
Download Solution PDFGiven
Area of Quadrilateral = 12√3 cm2
Concept Used
We know that The median of a triangle is cut the triangle in equal areas
Area of triangle = (√3/4) (side)2
Calculation
⇒ Area of ΔABC = Area of Quadrilateral BDGF × 3
⇒ Area of ΔABC = 12√3 × 3
⇒Area of ΔABC = 36√3 cm2
⇒ 36√3 = (√3/4) × (side)2
⇒ side = 12 cm
∴ the side of ΔABC is 12 cm
The side of an equilateral triangle is 12 cm. What is the radius of the circle circumscribing this equilateral triangle?
Answer (Detailed Solution Below)
Triangle Question 13 Detailed Solution
Download Solution PDFGiven:
The side of an equilateral triangle is 12 cm.
Concept used:
The radius of a circle circumscribing an equilateral triangle = side/√3
Calculation:
According to the concept,
Radius of the circle = 12/√3
⇒ (4 × 3)/√3
∵ 3 = √3 × √3 = (√3)2
⇒ 4 ×(√3)2/√3
⇒ 4√3
∴ The radius of the circle circumscribing this equilateral triangle is 4√3 cm.
The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
Answer (Detailed Solution Below)
Triangle Question 14 Detailed Solution
Download Solution PDFGiven data:
CIrcumradius = 3.2 cm
Calculations:
From the equilateral triangle property, O is the circumcenter as well as the centroid.
∴ OD =
⇒ AD = 3 × OD
⇒ AD = 3 × 3.2 = 9.6 cm
∴ The length of the altitudes of the equilateral triangle is 9.6 cm.
The altitude AD of a triangle ABC is 9 cm. If AB = 6√3 cm and CD = 3√3 cm, then what will be the measure of ∠A?
Answer (Detailed Solution Below)
Triangle Question 15 Detailed Solution
Download Solution PDFShortcut TrickWe know that the height of an equilateral triangle = a√3/2
Here, Height = 9 = 6√3 × √3/2
a = 6√3, so Height = a√3/2.
∴ Given triangle is an equilateral triangle.
So, ∠A = 60°.
Traditional method:
Concept:
Pythagoras theorem:
(AB)2 = (BD)2 + (AD)2
Equilateral Triangle:
AB = BC = AC
∠A = ∠B = ∠C = 60°
Calculation:
(6√3)2 = (BD)2 + 92
⇒ BD = 3√3 cm
∵ DC = BD = 3√3 cm
∴ BC = AC = AB = 6√3 cm &
∠A = ∠B = ∠C = 60°
∴ ABC is an equilateral triangle.