Chapter 6 · Class 10 CBSE · MCQ Test
Triangles MCQ Test — Class 10 CBSE
Practice 10 multiple-choice questions with instant answer reveal and explanations.
Triangles — MCQ Questions
1Consider a triangle ABC. A line DE is drawn parallel to BC, intersecting AB at D and AC at E. Which of the following statements correctly represents the Basic Proportionality Theorem (BPT)?
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Answer: AD/DB = AE/EC
Hint: BPT relates the ratios of the segments created on the sides of the triangle by the parallel line.
Solution:
The Basic Proportionality Theorem (BPT), also known as Thales Theorem, states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
In ΔABC, if DE || BC, then according to BPT, the ratio of the segments on AB is equal to the ratio of the segments on AC. — AD/DB = AE/EC
2In triangle PQR, points S and T are on PQ and PR respectively. If PS = 3 cm, SQ = 6 cm, PT = 4 cm, and TR = 8 cm, which of the following statements is true?
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Answer: ST || QR
Hint: Check if the ratios of the segments on the sides are equal.
Solution:
First, calculate the ratios of the segments on sides PQ and PR.
PS/SQ = 3 cm / 6 cm = 1/2.
PT/TR = 4 cm / 8 cm = 1/2.
Since PS/SQ = PT/TR (both are 1/2), by the converse of the Basic Proportionality Theorem, the line segment ST must be parallel to QR.
3Consider two triangles, ΔABC and ΔDEF. If ∠A = 50°, ∠B = 60°, ∠D = 50°, and ∠F = 70°, which similarity criterion would prove ΔABC ~ ΔDEF?
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Answer: AA Similarity
Hint: First, find the missing angles in both triangles using the angle sum property. Then compare the angles.
Solution:
In ΔABC: ∠C = 180° - (∠A + ∠B) = 180° - (50° + 60°) = 180° - 110° = 70°.
In ΔDEF: ∠E = 180° - (∠D + ∠F) = 180° - (50° + 70°) = 180° - 120° = 60°.
Now, we have: ∠A = ∠D = 50°, ∠B = ∠E = 60°, and ∠C = ∠F = 70°.
Since two corresponding angles (and thus all three) are equal, the triangles are similar by the AA (Angle-Angle) similarity criterion.
4Two triangles, ΔPQR and ΔXYZ, have sides such that PQ = 6 cm, QR = 8 cm, PR = 10 cm and XY = 3 cm, YZ = 4 cm, XZ = 5 cm. Which of the following statements is true about these triangles?
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Answer: They are similar by SSS criterion.
Hint: Check the ratios of corresponding sides. Ensure you match the smallest side with the smallest, medium with medium, and largest with largest.
Solution:
Let's check the ratios of corresponding sides:
PQ/XY = 6/3 = 2.
QR/YZ = 8/4 = 2.
PR/XZ = 10/5 = 2.
Since the ratios of all three pairs of corresponding sides are equal (PQ/XY = QR/YZ = PR/XZ = 2), the triangles ΔPQR and ΔXYZ are similar by the SSS (Side-Side-Side) similarity criterion.
5For two triangles ΔLMN and ΔRST to be similar by the SAS (Side-Angle-Side) criterion, which of the following sets of conditions is sufficient?
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Answer: LM/RS = MN/ST and ∠M = ∠S
Hint: Remember that for SAS similarity, the angle must be *included* between the two proportional sides.
Solution:
The SAS similarity criterion states that if two sides and the *included angle* of one triangle are proportional to two sides and the included angle of another triangle, then the two triangles are similar.
In option A, the sides LM and MN are proportional to RS and ST, and the angle ∠M is the included angle between LM and MN, while ∠S is the included angle between RS and ST. This matches the SAS criterion.
Options B, C, and D do not have the angle included between the proportional sides, hence they are incorrect.
6If two triangles, ΔABC and ΔPQR, are similar, then their corresponding angles are equal, and the ratio of their corresponding sides is _________.
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Answer: equal
Hint: Think about the definition of similar figures and how their sides relate.
Solution:
By the definition of similar triangles, two triangles are similar if their corresponding angles are equal AND their corresponding sides are in the same ratio.
This means that the ratio of their corresponding sides is equal. For example, if ΔABC ~ ΔPQR, then AB/PQ = BC/QR = AC/PR.
7Which of the following conditions is essential for applying the Pythagoras Theorem to a triangle?
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Answer: The triangle must be a right-angled triangle.
Hint: Recall the specific type of triangle for which the theorem a² + b² = c² holds true.
Solution:
The Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This theorem is applicable *only* to right-angled triangles.
8Ravi was trying to find the length of the diagonal of a square with side length 5 cm. He reasoned: 'Let the diagonal be d. By Pythagoras theorem, d² = 5² + 5², so d² = 25 + 25 = 50. Therefore, d = √50 ≈ 7.07 cm.' What is the error in Ravi's reasoning, if any?
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Answer: There is no error in Ravi's reasoning; his solution is correct.
Hint: A square's diagonal divides it into two right-angled triangles. Check if the application of the theorem and the calculation are sound.
Solution:
A square has four right angles. When a diagonal is drawn, it forms two right-angled triangles.
The sides of these right-angled triangles are the sides of the square (5 cm), and the diagonal is the hypotenuse.
Ravi correctly applied the Pythagoras Theorem: d² = 5² + 5².
He correctly calculated d² = 25 + 25 = 50, and d = √50 ≈ 7.07 cm. His reasoning and calculation are sound.
9A triangle has side lengths 7 cm, 24 cm, and 25 cm. Based on these lengths, what type of triangle is it?
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Answer: A right-angled triangle
Hint: Use the converse of the Pythagoras Theorem. Check if the square of the longest side equals the sum of the squares of the other two sides.
Solution:
Let the sides be a = 7 cm, b = 24 cm, and c = 25 cm. The longest side is c = 25 cm.
Calculate the sum of the squares of the two shorter sides: a² + b² = 7² + 24² = 49 + 576 = 625.
Calculate the square of the longest side: c² = 25² = 625.
Since a² + b² = c² (625 = 625), by the converse of the Pythagoras Theorem, the triangle is a right-angled triangle.
10If two triangles are similar, and the ratio of their corresponding sides is 2:3, what is the ratio of their areas?
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Answer: 4:9
Hint: Recall the theorem that relates the ratio of areas of similar triangles to the ratio of their corresponding sides.
Solution:
The theorem on areas of similar triangles states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Given the ratio of corresponding sides = 2:3.
Ratio of areas = (Ratio of sides)² = (2/3)² = 4/9.
So, the ratio of their areas is 4:9.
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Tips for Triangles MCQs
- 1Read each question carefully and identify what is being asked before looking at the options.
- 2Try to solve the problem mentally or on paper first, then match your answer with the options.
- 3Use elimination — rule out clearly wrong options to improve your chances even when unsure.
- 4Check units, signs, and edge cases — these are common traps in Triangles MCQs.
- 5Review your mistakes after completing the test to build lasting understanding.
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