Chapter 6 · Triangles

NCERT Exemplar Class 10 Maths Exercise 6.2

All 12 questions of NCERT Exemplar Class 10 Maths Exercise 6.2 (Triangles), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Is ​the ​triangle with ​sides 25 cm, 5 cm ​and 24 ​cm a ​right triangle? ​Give reasons for your answer.
   Show answer

   No. Since $25^2=625\ne 5^2+24^2=601$, the triangle is not a right triangle.

   Step-by-step solution →
2. Q2
   It is given that triangle DEF ~ triangle RPQ. ​Is it ​true to say ​that angle D = ​angle R and angle ​F = ​angle P? Why?
   Show answer

   Only partly true: angle D = angle R is correct, but angle F = angle P is wrong. From DEF ~ RPQ we get angle F = angle Q (and angle E = angle P).

   Step-by-step solution →
3. Q3
   A and ​B are respectively the points on the sides PQ and PR ​of a triangle PQR ​such that $PQ=12.5$ ​cm, $PA=5$ cm, $BR=6$ cm and $PB=4$ cm. ​Is $AB\parallel QR$? ​Give reasons for your answer.
   Show answer

   Yes, AB is parallel to QR. Since PA/AQ = 5/7.5 = 2/3 and PB/BR = 4/6 = 2/3 are equal, by the converse of the Basic Proportionality Theorem AB || QR.

   Step-by-step solution →
4. Q4
   In Fig. ​6.4, BD ​and ​CE ​intersect each ​other at the ​point ​P. Is $△PBC\sim △PDE$? Why?

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   Yes, $△PBC\sim △PDE$ by the SAS similarity criterion: $\frac{PB}{PD}=\frac{PC}{PE}=\frac{1}{2}$ and $\angle BPC=\angle DPE$ (vertically opposite).

   Step-by-step solution →
5. Q5
   In triangles $PQR$ and $MST$, $\angle P=55^{\circ }$, $\angle Q=25^{\circ }$, $\angle M=100^{\circ }$ ​and $\angle S=25^{\circ }$. Is $△QPR\sim △TSM$? ​Why?
   Show answer

   No, $△QPR\sim △TSM$ as written is not correct: $\angle Q=25^{\circ }\ne \angle T=55^{\circ }$. The two triangles do have equal angle sets $\{25^{\circ },55^{\circ },100^{\circ }\}$, so they are similar, but in the correct correspondence $△QPR\sim △STM$.

   Step-by-step solution →
6. Q6
   Is the ​following ​statement true? ​Why? "Two quadrilaterals are similar, if ​their ​corresponding angles ​are ​equal".
   Show answer

   No, the statement is false. A square and a non-square rectangle have equal corresponding angles but unequal side ratios, so they are not similar.

   Step-by-step solution →
7. Q7
   Two sides and the perimeter of one triangle are respectively ​three times ​the ​corresponding sides and the perimeter of ​the other triangle. ​Are the two triangles ​similar? Why?
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   Yes, the triangles are similar by the SSS criterion, since the third sides are also in ratio 3:1.

   Step-by-step solution →
8. Q8
   If in two right triangles, ​one of ​the acute ​angles of one triangle is equal to ​an ​acute angle of the ​other triangle, can you say ​that the two triangles will be similar? Why?
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   Yes, the two right triangles are similar. The right angles are equal, one pair of acute angles is equal (given), so by the angle sum property the third pair is also equal — the triangles are equiangular and hence similar by the AAA (AA) similarity criterion.

   Step-by-step solution →
9. Q9
   The ratio of the corresponding altitudes of two similar triangles is $\frac{3}{5}$. Is it ​correct ​to say ​that ratio ​of ​their areas is $\frac{6}{5}$? Why?
   Show answer

   No, it is not correct. The ratio of areas is ${\left(\frac{3}{5}\right)}^2=\frac{9}{25}$, not $\frac{6}{5}$.

   Step-by-step solution →
10. Q10
    D ​is a point ​on side QR ​of $△PQR$ such that $PD\perp QR$. Will it be ​correct to say that $△PQD\sim △RPD$? ​Why?
    Show answer

    No. $△PQD\sim △RPD$ only when $\angle QPR=90^{\circ }$, and that is not given.

    Step-by-step solution →
11. Q11
    In ​Fig. 6.5, if $\angle D=\angle C$, ​then ​is it ​true that $△ADE\sim △ACB$? ​Why?

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    Yes, $△ADE\sim △ACB$ by the AA similarity criterion, since $\angle A$ is common and $\angle D=\angle C$.

    Step-by-step solution →
12. Q12
    Is it ​true to ​say that if in two triangles, an angle of one triangle is equal to an angle of another triangle ​and two ​sides ​of one ​triangle are proportional to the two sides of the other ​triangle, then the triangles are similar? Give reasons for your answer.
    Show answer

    No, it is not always true. The triangles are similar only if the equal angle is the angle **included** between the two proportional sides (SAS similarity criterion). If the equal angle is not the included angle, similarity is not guaranteed.

    Step-by-step solution →