A Tale of Three Intersecting Lines MCQs for UPSC Prelims
110 practice questions covering A Tale of Three Intersecting Lines, organised into 1 topics.
110 questions include a written explanation.
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Which set of information is not enough to fix one particular triangle?
AThe three sides
BThe three angles only
CTwo sides and the angle between them
DTwo angles and the side between them
Show answer and explanation
Correct answer: B - The three angles only
Three angles fix the shape but leave the size open, so many triangles of different sizes share them. Three sides, two sides with the angle between them, and two angles with the side between them each pin down a single triangle.
Q2
hard
In triangle $ABC$, $\angle C = 80^\circ$ and $\angle A$ is $20^\circ$ more than $\angle B$. What is $\angle B$?
A$60^\circ$
B$30^\circ$
C$40^\circ$
D$50^\circ$
Show answer and explanation
Correct answer: C - $40^\circ$
The angles $\angle A$ and $\angle B$ share $180^\circ - 80^\circ = 100^\circ$. Removing the extra $20^\circ$ leaves $80^\circ$ to be split equally, so $\angle B = 40^\circ$ and $\angle A = 60^\circ$. The value $60^\circ$ is $\angle A$, $50^\circ$ ignores the $20^\circ$ difference, and $30^\circ$ takes $20^\circ$ away from $50^\circ$.
Q3
easy
A triangle has sides $6\ \text{cm}$, $6\ \text{cm}$ and $9\ \text{cm}$. What type of triangle is it by its sides?
ANo triangle is possible with these lengths
BScalene
CIsosceles
DEquilateral
Show answer and explanation
Correct answer: C - Isosceles
Exactly two sides are equal, which makes the triangle isosceles. Equilateral needs all three sides equal, and scalene needs all three different. Since $6 + 6 = 12$ is more than $9$, the triangle can certainly be drawn.
Q4
medium
Assertion (A): In an obtuse-angled triangle, two of the three altitudes lie outside the triangle. Reason (R): The three angles of any triangle add up to $180^\circ$.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: B - Both A and R are true, but R is not the correct explanation of A
The two vertices with acute angles each send a perpendicular that lands beyond the opposite side, so two altitudes fall outside and A is true. R is also true, as the angle sum of a triangle is always $180^\circ$. That sum holds for acute-angled triangles too, whose altitudes all stay inside, so R does not explain A.
Q5
medium
What is true of the three altitudes of an equilateral triangle?
AThey are all equal in length
BOnly two of them are equal
CEach of them equals a side of the triangle
DThey all have different lengths
Show answer and explanation
Correct answer: A - They are all equal in length
An equilateral triangle has three equal sides and three equal angles, so its three altitudes match one another as well. Each altitude is shorter than a side, since it is a perpendicular distance rather than a full side. With this much symmetry, no altitude can differ from the other two.
Q6
medium
Which set of angles belongs to a right-angled triangle?
A$60^\circ, 60^\circ, 60^\circ$
B$100^\circ, 40^\circ, 40^\circ$
C$45^\circ, 45^\circ, 45^\circ$
D$25^\circ, 65^\circ, 90^\circ$
Show answer and explanation
Correct answer: D - $25^\circ, 65^\circ, 90^\circ$
The first set contains an angle of exactly $90^\circ$ and adds to $180^\circ$, so it is right-angled. Three angles of $45^\circ$ total only $135^\circ$, so they form no triangle. Three angles of $60^\circ$ give an acute-angled triangle, and the last set is obtuse-angled.
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