True False[1 Marks ]

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:
BY conver of pythagoras theoerm, this $\triangle$ will be right angle triangle if
$(25)^2= (5)^2 + (24)^2$
$^{\Rightarrow 625 = 25 + 576}$
$\Rightarrow625\neq601 $
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

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 51 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 61 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 71 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 81 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 91 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 101 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 111 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 121 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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