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Maths True False[1 Marks ]

Triangles · Jharkhand Board (JAC) STD 10 (Jharkhand - English Medium). 18 questions.

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True False[1 Marks ]

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18 questions · self-marked practice — reveal the answer and mark yourself.

Question 11 Mark
A and B are respectively the points on the sides PQ and PR of a triangle PQR such that PQ = 12.5cm, PA = 5cm, BR= 6cm and PB = 4cm. Is AB || QR? Give reasons for your answer.
Answer
True: By converse of BPT, AB will be parallel to QR if AB, divides PQ and PR in the same ratio i.e.,
$\frac{\text{AP}}{\text{AQ}}=\frac{\text{PB}}{\text{BR}}$
$\Rightarrow\frac{5}{12.5-5}=\frac{4}{6}$
$\Rightarrow\frac{5.0}{7.5}=\frac{2}{3}$
$\Rightarrow\frac{2}{3}=\frac{2}{3}$
So, AB is parallel to QR. Hence, the given statement AB || QR is true.
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Question 21 Mark
Is the triangle with sides 25cm, 5cm and 24cm a right triangle? Give reasons for your answer.
Answer
False: $B Y$ conver of pythagoras theoerm, this $\triangle$ will be right angle triangle if $(25) 2=(5) 2+(24) 2 \Rightarrow 625=25+576$ $\Rightarrow 625 \neq 601$
So, the given triangle is not right angled triangle.
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Question 31 Mark
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?
Answer
True: In $\triangle\text{ABC}$ and $\triangle\text{PQR}$ $\angle{\text{B}}=\angle{\text{Q}}=90^\circ$ [Given]
$\angle{\text{C}}=\angle{\text{R}}$ [Given]
$\therefore\triangle\text{ABC}\sim\triangle\text{PQR}$ [by AA similarity criterion]
Hewnce, the statement that two triangles are similar is true.
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Question 41 Mark
Write the truth value (T/F) of the following statements:
Two polygons are similar, if their corresponding sides are proportional.
Answer
False.
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Question 51 Mark
The ratio of the corresponding altitudes of two similar triangles is $\frac{3}{5}.$ Is it correct to say that ratio of thier areas is $\frac{6}{5}?$ Why?
Answer
False:
If two triangles are similar, then the ratio of areas of two triangles will be equal to the square of the ratios of their corresponding sides or altitudes or angle bisectors,
$\text{If }\triangle\text{ABC}\sim\triangle\text{PQR},\text{ then}$
$\frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{PQR})}=\bigg(\frac{\text{AD}}{\text{PM}}\bigg)^2$
$\Rightarrow\frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{PQR})}=\Big(\frac{3}{5}\Big)^2$
$=\frac{9}{25}\neq\frac{6}{5}$
So, the given statement is false.
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Question 61 Mark
Write the truth value (T/F) of the following statements:
Any two congruent figures are similar.
Answer
True.
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Question 71 Mark
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?
Answer
True: Let the two sides of $\triangle\text{ABC}$ are AB = 3cm, AC = 4cm and perimeter AB + BC + AC = 13cm, then BC = 13 - 7 = 6cm According to the question, the sides of another $\triangle\text{DEF}$ are DE = 3 × 3 = 9 DE = 3 × 4 = 12 and DE + DF + EF = 3 × 13 = 39 So, EF = 39 - 12 - 9 = 18$\therefore\frac{\text{DE}}{\text{AB}}=\frac{9}{3}=\frac{3}{1}$
$\frac{\text{DF}}{\text{AC}}=\frac{12}{4}=\frac{3}{1}$
$\frac{\text{EF}}{\text{BC}}=\frac{18}{6}=\frac{3}{1}$
$\therefore\frac{\text{DE}}{\text{AB}}=\frac{\text{DF}}{\text{AC}}=\frac{\text{EF}}{\text{BC}}=\frac{3}{1}$
As the ratio of corresponding sides in two $\triangle\text{S}$ are same then $\triangle\text{DEF}\sim\triangle\text{ABC}$ by SSS similarity criterion. Hence, the triangles are similar or thr given statement is true.
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Question 81 Mark
Write the truth value (T/F) of the following statements:
Any two similar figures are congruent.
Answer
False.
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Question 91 Mark
In triangles PQR and MST, $\angle\text{P}=55^\circ,\angle\text{Q}=25^\circ\angle{\text{M}}=100^\circ$ and $\angle{\text{S}}=25^\circ.$ Is $\triangle\text{QPR}\sim\triangle\text{TSM}?$ Why?
Answer
False:
$\triangle\text{QPR}$ and $\triangle\text{TSM}$ will be similar if its corresponding angles are equal
$\angle\text{Q}=25^\circ$
$\angle\text{P}=55^\circ$
$\Rightarrow\angle\text{R}=180^\circ-(25^\circ+55^\circ)$
$\Rightarrow\angle{\text{R}}=180^\circ+80^\circ$
$\Rightarrow\angle{\text{R}}=100^\circ$
$\Rightarrow\angle{\text{S}}=25^\circ$
$\Rightarrow\angle{\text{M}}=100^\circ$
$\Rightarrow\angle\text{T}=180^\circ-(100^\circ+25^\circ)$
$\Rightarrow\angle\text{T}=55^\circ$
$\therefore\angle\text{Q}\neq\angle{\text{T}}$
$\angle\text{P}\neq\angle{\text{S}}$
$\angle\text{R}\neq\angle{\text{M}}$
So, $\triangle\text{QPR}$ is not similar to $\triangle\text{TSM}.$ So, the given statement $\triangle\text{QPR}\sim\triangle\text{TSM}$ is false.
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Question 101 Mark
Is the following statement true? Why? “Two quadrilaterals are similar, if their corresponding angles are equal”.
Answer
False:
Two quadriaterals will be similar if their corresoonding angles as well as ratio of sides are also equal so, the given statement is falase.
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Question 111 Mark
It is given that $\triangle\text{DEF}\sim\triangle\text{RPQ}.$ Is it true to say that $\angle\text{D}=\angle\text{R}\text{ and }\angle\text{F}=\angle\text{P}?$ Why?
Answer
False: When $\triangle\text{DEF}\sim\triangle\text{RPQ}$ each angle of a triangle will be equal to the corresponding angle of similar triangle so$\angle\text{D}=\angle\text{R}$
$\angle\text{E}=\angle\text{P}$
$\angle\text{F}=\angle\text{Q}$
So, $\angle\text{D}=\angle\text{R}$ is true but $\angle\text{F}\neq\angle\text{P.}$ Hence, it is not true that $\angle\text{D}=\angle\text{R}$ and $\angle\text{F}=\angle\text{P}.$
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Question 121 Mark
Write the truth value (T/F) of each of the following statements:
Two triangles are similar, if their corresponding sides are proportional.
Answer
True.
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Question 131 Mark
If $\angle\text{D}=\angle\text{C},$ then is it true that $\triangle\text{ADE}\sim\triangle\text{ACB}?$ Why?
Answer
True: In $\triangle\text{ADE}$ and $\triangle\text{ACB}$$\angle\text{D}=\angle\text{C}$ [Given]
$\angle\text{A}=\angle\text{A}$ [Common]
$\therefore\triangle\text{ADE}\sim\triangle\text{ACB}$ [by AA similarity criterion]
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Question 141 Mark
Write the truth value (T/F) of the following statements:
Two polygons are similar, if their corresponding angles are proportional.
Answer
False.
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Question 151 Mark
Write the truth value (T/F) of the following statements:
Two triangles are similar, if their corresponding angles are proportional.
Answer
True.
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Question 161 Mark
In BD and CE intersect each other at the point P. Is $\triangle\text{PBC}\sim\triangle\text{PDE}?$ Wht?

 
Answer

True:
If $\triangle\text{PBC}$ and $\triangle\text{PDE},$ we have
$\angle\text{BPC}=\angle\text{DPE}$ [Vertically opposite angles]
$\frac{\text{BP}}{\text{PD}}=\frac{5}{10}=\frac{1}{2}$
$\frac{\text{PC}}{\text{PE}}=\frac{6}{12}=\frac{1}{2}$
$\therefore\frac{\text{BP}}{\text{PD}}=\frac{\text{PC}}{\text{PE}}$
Hence, $\triangle\text{BPC}\sim\triangle\text{DPE}$ [by SAS similarity criterion]
Hence, the given statement is true.
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Question 171 Mark
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.
Answer
False:
Here, the ratio of two sides of a triangles is equal to the ratio of corresponding two sides of other triangle, although the one angle of one angle of othere triangles but, not included angles of proportional sides are equal.
So, triangles are not similar. Hence, the given statement is false.
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Question 181 Mark
D is a point on side QR of $\triangle\text{PQR}$ such that $\text{PD}\perp\text{QR}$ will it be correct to say that $\triangle\text{PQD}\sim\triangle\text{RPD}?$ Why?
Answer

False:
In $\triangle\text{PQD}$ and $\triangle\text{PDR},$
$\text{PD}\perp\text{QR}$
$\therefore\angle\text{PDQ}\sim\angle\text{PDR}=90^\circ$
PD does not bisect $\angle\text{P}.$
$\therefore\angle1\neq\angle2$
$\angle\text{Q}\neq\angle\text{R}\ [\because\text{PQ}\neq\text{QR}]$
Any ratio of sides are also not equal. So, $\triangle\text{PQD}$ is not similar to $\triangle\text{PDR}.$ Hence, the given statement is false.
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