Chapter 6 · Triangles

NCERT Exemplar Class 10 Maths Exercise 6.3

All 15 questions of NCERT Exemplar Class 10 Maths Exercise 6.3 (Triangles), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   In a triangle PQR, ​PR^2 ​- ​PQ^2 = QR^2 and M is ​a ​point on side PR such that QM is ​perpendicular to PR. Prove that ​QM^2 = PM x ​MR.
   Show answer

   $Q{M}^2=PM\times MR$, proved: the given relation makes angle Q a right angle, so QM is the altitude on the hypotenuse and the similar triangles PMQ and QMR give the geometric-mean relation.

   Step-by-step solution →
2. Q2
   Find the ​value ​of $x$ ​for ​which $DE\parallel AB$ ​in Fig. 6.8.

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   $x=2$

   Step-by-step solution →
3. Q3
   In ​Fig. ​6.9, if $\angle 1=\angle 2$ ​and $△NSQ\cong △MTR$, ​then prove ​that $△PTS\sim △PRQ$.

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   Proved: $△PTS\sim △PRQ$. Chain of reasoning: $SQ=TR$ (CPCT from the given congruence) and $PS=PT$ (sides opposite the equal angles $\angle 1$, $\angle 2$), so $\frac{PS}{SQ}=\frac{PT}{TR}$, giving $ST\parallel QR$ by the converse of the Basic Proportionality Theorem; the resulting corresponding angles together with the common angle $P$ make the two triangles equiangular (AAA).

   Step-by-step solution →
4. Q4
   Diagonals of a trapezium ​PQRS ​intersect each ​other at ​the point O, PQ $\parallel$ RS ​and PQ = ​3 RS. Find the ratio of the areas of triangles POQ and ROS.
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   The ratio of the areas of triangles POQ and ROS is $9:1$.

   Step-by-step solution →
5. Q5
   In Fig. 6.10, ​if $AB\parallel DC$ ​and AC and PQ ​intersect ​each other at ​the point ​O, prove that $OA\cdot CQ=OC\cdot AP$.

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   Proved: $AB\parallel DC$ gives $△OAP\sim △OCQ$ (AA), so $\frac{OA}{OC}=\frac{AP}{CQ}$, hence $OA\cdot CQ=OC\cdot AP$.

   Step-by-step solution →
6. Q6
   Find ​the ​altitude of ​an ​equilateral ​triangle of side 8 ​cm.
   Show answer

   The altitude of the equilateral triangle is $4\sqrt{3}$ cm, approximately 6.93 cm.

   Step-by-step solution →
7. Q7
   If $△ABC\sim △DEF$, $AB=4$ cm, $DE=6$ ​cm, $EF=9$ cm ​and $FD=12$ ​cm, ​find the ​perimeter of $△ABC$.
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   The perimeter of $△ABC$ is $18$ cm.

   Step-by-step solution →
8. Q8
   In ​Fig. 6.11, if ​DE $\parallel$ BC, find ​the ​ratio of ​ar (ADE) and ​ar ​(DECB).

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   $ar(△ADE):ar(DECB)=1:3$.

   Step-by-step solution →
9. Q9
   ABCD is a ​trapezium ​in which $AB\parallel DC$ and P and Q are ​points on AD and BC, ​respectively such that $PQ\parallel DC$. ​If $PD=18$ ​cm, $BQ=35$ cm ​and $QC=15$ cm, find AD.
   Show answer

   $AD=60$ cm.

   Step-by-step solution →
10. Q10
    Corresponding sides of two ​similar triangles ​are in the ratio ​of $2:3$. ​If ​the area of the smaller triangle is $48$ cm${}^2$, find ​the area of ​the larger triangle.
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    The area of the larger triangle is $108$ cm${}^2$.

    Step-by-step solution →
11. Q11
    In a ​triangle ​PQR, N ​is a point ​on ​PR such that QN is perpendicular to ​PR. If PN.NR = QN^2, ​prove that ​angle PQR = 90 degrees.
    Show answer

    $\angle PQR=90^{\circ }$, proved using SAS similarity of triangles PNQ and QNR.

    Step-by-step solution →
12. Q12
    Areas of two similar triangles ​are ​36 cm${}^2$ ​and 100 cm${}^2$. If the length of a side of ​the larger triangle is 20 cm, find the length ​of the corresponding side ​of the smaller triangle.
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    The corresponding side of the smaller triangle is $12$ cm.

    Step-by-step solution →
13. Q13
    In ​Fig. ​6.12, ​if $\angle ACB=\angle CDA$, $AC=8$ cm ​and $AD=3$ ​cm, find ​BD.

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    BD = 55/3 cm, approximately 18.33 cm.

    Step-by-step solution →
14. Q14
    A 15 metres high tower casts a shadow 24 metres long ​at a certain time ​and at the same time, a telephone pole casts ​a ​shadow 16 metres long. Find ​the ​height of the telephone pole.
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    The telephone pole is $10$ metres high.

    Step-by-step solution →
15. Q15
    Foot of a 10 m ​long ladder leaning against a vertical wall is 6 m ​away ​from the base of the wall. Find ​the height of ​the ​point on ​the wall where the top of ​the ladder reaches.
    Show answer

    The top of the ladder reaches a height of $8$ m on the wall.

    Step-by-step solution →