Class 10 Maths · Chapter 6

Triangles

424 Class 10 Maths questions on Triangles from the NCERT textbook, NCERT Exemplar and CBSE board papers 2018–2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 12 of 22

1. Q221 CBSE 2025
   Sides $AB$ and $BC$ ​and median $AD$ ​of ​triangle $ABC$ are respectively proportional to sides $PQ$ and $QR$ and ​median $PM$ of $△PQR$. ​Show that $△ABC\sim △PQR$.
   Show answer

   $△ABC\sim △PQR$, proved by SSS similarity of $△ABD$ and $△PQM$ (which gives $\angle B=\angle Q$) followed by SAS similarity.

   Step-by-step solution →
2. Q222 CBSE 2025
   In $△ABC$, if $AD\perp BC$ and $A{D}^2=BD\times DC$, ​then ​prove ​that $\angle BAC=90^{\circ }$.
   Show answer

   Hence $\angle BAC=90^{\circ }$, as required: $△DBA\sim △DAC$ by SAS similarity gives $\angle BAC=\angle B+\angle C$, and the angle sum of $△ABC$ then gives $2\angle BAC=180^{\circ }$.

   Step-by-step solution →
3. Q223 CBSE 2025
   In ​a $△ABC$, $P$ and $Q$ ​are points ​on $AB$ and $AC$ ​respectively such that $PQ\parallel BC$. Prove ​that the median $AD$, ​drawn from $A$ to $BC$, bisects $PQ$.

   Show answer

   Let $AD$ meet $PQ$ at $M$. Since $△APM\sim △ABD$ and $△AMQ\sim △ADC$, $\frac{PM}{BD}=\frac{AM}{AD}=\frac{MQ}{DC}$; as $BD=DC$, we get $PM=MQ$. Hence the median $AD$ bisects $PQ$.

   Step-by-step solution →
4. Q224 CBSE 2025
   A triangular window of a building is shown above. ​Its diagram represents ​a $△ABC$ ​with $\angle A=90^{\circ }$ and $AB=AC$. Points $P$ and $R$ ​trisect $AB$ and $PQ\parallel RS\parallel AC$. Based ​on ​the ​above, answer the ​following questions.

   1. iShow that $△BPQ\sim △BAC$.[1]
   2. iiProve that $PQ=\frac{1}{3}AC$.[1]
   3. iii(a)If $AB=3$ m, find length $BQ$ and $BS$. Verify that $BQ=\frac{1}{2}BS$.[2]
   4. iii(b)Prove that $B{R}^2+R{S}^2=\frac{4}{9}B{C}^2$.[2]
   Show answer

   (i) $\angle B$ is common and $\angle BPQ=\angle BAC$ (corresponding angles, $PQ\parallel AC$), so $△BPQ\sim △BAC$ by AA similarity; (ii) $\frac{PQ}{AC}=\frac{BP}{BA}=\frac{1}{3}$, so $PQ=\frac{1}{3}AC$; (iii)(a) $BQ=\sqrt{2}$ m and $BS=2\sqrt{2}$ m, and $\frac{1}{2}BS=\sqrt{2}$ m $=BQ$; (iii)(b) $B{R}^2+R{S}^2=\frac{4}{9}\left(A{B}^2+A{C}^2\right)=\frac{4}{9}B{C}^2$. Hence proved.

   Step-by-step solution →
5. Q225 CBSE 2025
   Raghu is trying to find the height of a tower ​near his house, using the properties of similar triangles. The height of Raghu's house ​is $12$ m. When Raghu's ​house casts a shadow $6$ m long on the ground, ​the tower ​casts a shadow $40$ m long on ​the ground. At the same time, the house of his friend Ramesh casts $12$ m ​long shadow on the ground. Based on the above information, answer the following:

   1. (i)What is the height of the tower?[1]
   2. (ii)What is the height of Ramesh's house?[1]
   3. (iii)(a)When the tower casts a shadow of $60$ m long, what will be the length of shadow of Ramesh's house?[2]
   4. (iii)(b) ORWhen the tower casts a shadow of $48$ m long, what will be the length of shadow of Raghu's house?[2]
   Show answer

   (i) The height of the tower is $80$ m; (ii) The height of Ramesh's house is $24$ m; (iii)(a) The shadow of Ramesh's house is $18$ m long; OR (iii)(b) The shadow of Raghu's house is $7.2$ m long.

   Step-by-step solution →
6. Q226 CBSE 2025
   Case Study - 2: ​On a ​road leading to ​a school, there is a triangle (ABC) shaped board on the ​road. It is divided into two parts by a line $DE$, which is parallel to $BC$. ​On the upper ​part it is written 'DRIVE SLOW' and ​on the lower part it is written, 'SCHOOL AHEAD'. Based ​on the information given above, answer the following questions:

   1. iIf $AD=3$ cm, $BD=5$ cm and $AE=4$ cm, then find the length of $AC$.[1]
   2. iiIf $\angle ADE=50^{\circ }$ and $\angle DAE=45^{\circ }$, then find $\angle ACB$.[1]
   3. iii(a)If $AD=4$ cm and $BD=6$ cm, then find $\frac{DE}{BC}$.[2]
   4. iii(b)If $AD=3$ cm, $BD=6$ cm and $AE=5$ cm, then find $\frac{AB}{AC}$.[2]
   Show answer

   (i) $AC=\frac{32}{3}$ cm; (ii) $\angle ACB=85^{\circ }$; (iii)(a) $\frac{DE}{BC}=\frac{2}{5}$; (iii)(b) $\frac{AB}{AC}=\frac{3}{5}$

   Step-by-step solution →
7. Q227 CBSE 2024
   In the ​given ​figure, ​if in $△ABC$, $DE\parallel BC$, ​then which ​of ​the following ​equality holds?

   1. (a)$\frac{AD}{AB}=\frac{AE}{CE}$
   2. (b)$\frac{AD}{AB}=\frac{AE}{AC}$
   3. (c)$\frac{AD}{BD}=\frac{AE}{AC}$
   4. (d)$\frac{AD}{AB}=\frac{AC}{AE}$
   Show answer

   (b) $\frac{AD}{AB}=\frac{AE}{AC}$

   Step-by-step solution →
8. Q228 CBSE 2024
   In the given figure, if M ​and N are points on ​the sides OP and OS respectively ​of $△OPS$, such that $MN\parallel PS$, then ​the length of ​OP ​is ​:

   1. (a)$6.8$ cm
   2. (b)$17$ cm
   3. (c)$15.3$ cm
   4. (d)$9.6$ cm
   Show answer

   (c) $15.3$ cm

   Step-by-step solution →
9. Q229 CBSE 2024
   If ​in the given figure, $DE\parallel BC$. ​If $AD=2.8$ cm, $DB=2.1$ cm ​and $EC=4.8$ cm, ​then ​the value of $x$ ​is :

   1. (a)$3.6$ cm
   2. (b)$2.4$ cm
   3. (c)$6.4$ cm
   4. (d)$4.8$ cm
   Show answer

   (c) $6.4$ cm

   Step-by-step solution →
10. Q230 CBSE 2024
    In the given figure $△ABC$ ​is shown. $DE$ ​is parallel to $BC$. ​If $AD=5$ cm, $DB=2.5$ cm ​and $BC=12$ cm, then $DE$ ​is ​equal to

    1. (a)$10$ cm
    2. (b)$6$ cm
    3. (c)$8$ cm
    4. (d)$7.5$ cm
    Show answer

    (c) $8$ cm

    Step-by-step solution →
11. Q231 CBSE 2024
    In a ​right-angled triangle ​ABC, $\angle A=90^{\circ }$ and $AB=AC$. ​The value of $\sin C$ ​is ​:

    1. (a)$0$
    2. (b)$\frac{\sqrt{3}}{2}$
    3. (c)$\frac{1}{2}$
    4. (d)$\frac{1}{\sqrt{2}}$
    Show answer

    (d) $\frac{1}{\sqrt{2}}$

    Step-by-step solution →
12. Q232 CBSE 2024
    If ​in ​triangles ​ABC ​and PQR, $\frac{AB}{QR}=\frac{BC}{PR}$, then ​they will be ​similar, when :

    1. (a)$\angle B=\angle Q$
    2. (b)$\angle A=\angle R$
    3. (c)$\angle B=\angle R$
    4. (d)$\angle C=\angle Q$
    Show answer

    (c) $\angle B=\angle R$

    Step-by-step solution →
13. Q233 CBSE 2024
    A line $l$ intersects ​the sides ​PQ and PR ​of a $△PQR$ at L and M respectively ​such that $LM\parallel QR$. If $PL=5.7$ cm, $PQ=15.2$ cm ​and $MR=5.5$ cm, ​then the length ​of PM ​(in cm) is :

    1. (a)$3$
    2. (b)$1.8$
    3. (c)$2.5$
    4. (d)$3.3$
    Show answer

    (d) $3.3$

    Step-by-step solution →
14. Q234 CBSE 2024
    The perimeters ​of two similar triangles $ABC$ and $PQR$ are $56$ ​cm and $48$ ​cm ​respectively. $PQ/AB$ ​is ​equal to

    1. (a)$\frac{7}{8}$
    2. (b)$\frac{6}{7}$
    3. (c)$\frac{7}{6}$
    4. (d)$\frac{8}{7}$
    Show answer

    (b) $\frac{6}{7}$

    Step-by-step solution →
15. Q235 CBSE 2024
    If ​the diagonals of ​a quadrilateral ​divide each other ​proportionally, then it ​is a :

    1. (a)parallelogram
    2. (b)rectangle
    3. (c)square
    4. (d)trapezium
    Show answer

    (d) trapezium

    Step-by-step solution →
16. Q236 CBSE 2024
    In $△ABC$, $DE\parallel BC$ (as ​shown ​in ​the ​figure). ​If $AD=2$ ​cm, $BD=3$ cm, $BC=7.5$ cm, ​then the ​length of $DE$ (in cm) is :

    1. (a)2.5
    2. (b)3
    3. (c)5
    4. (d)6
    Show answer

    (b) 3

    Step-by-step solution →
17. Q237 CBSE 2024
    In $△ABC$, $DE\parallel BC$ (as ​shown in the figure). If $AD=4$ ​cm, $AB=9$ ​cm and $AC=13.5$ ​cm, ​then the ​length of $EC$ ​is :

    1. (a)6 cm
    2. (b)7.5 cm
    3. (c)9 cm
    4. (d)5.7 cm
    Show answer

    (b) 7.5 cm

    Step-by-step solution →
18. Q238 CBSE 2024
    Which of ​the ​following ​is **not** the criterion for ​similarity of ​two triangles?

    1. (a)SAS
    2. (b)SSS
    3. (c)AAA
    4. (d)RHS
    Show answer

    (d) RHS

    Step-by-step solution →
19. Q239 CBSE 2024
    In two $△$s ​ABC ​and ​PQR, ​if $\frac{AB}{QR}=\frac{BC}{QP}=\frac{AC}{PR}$, ​then:

    1. (a)$△PQR\sim △CAB$
    2. (b)$△PQR\sim △ABC$
    3. (c)$△PQR\sim △CBA$
    4. (d)$△PQR\sim △BCA$
    Show answer

    (c) $△PQR\sim △CBA$

    Step-by-step solution →
20. Q240 CBSE 2024
    In the given figure, in $△ABC$, $DE\parallel BC$. If $AD=2.4$ ​cm, $DB=4$ ​cm and $AE=2$ cm, ​then the ​length of $AC$ is ​:

    1. (a)$\frac{10}{3}$ cm
    2. (b)$\frac{3}{10}$ cm
    3. (c)$\frac{16}{3}$ cm
    4. (d)$1.2$ cm
    Show answer

    (c) $\frac{16}{3}$ cm

    Step-by-step solution →