Triangles Test

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Triangles Test - 24

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Weekly Quiz Competition

Question 1
1 / -0

In the figure, $$BC\parallel DE$$ and $$\dfrac {AB}{AD} = \dfrac {3}{4}$$
What is the ratio between the area of $$\triangle ABC$$ and $$\triangle ADE$$?

A
$$\dfrac34$$

B
$$\dfrac9{32}$$

C
$$\dfrac38$$

D
$$\dfrac{9}{16}$$

E
$$\dfrac{8}{9}$$

Solution

GIven: $$BC\parallel DE$$ & $$\cfrac{AB}{AD}=\cfrac{3}{4}$$
In $$\triangle ABC$$ & $$\triangle ADE$$
$$\triangle ABC\sim \triangle ADE$$ (by $$AAA$$ property)
$$\therefore {(\cfrac{AB}{AD})}^{2}=\cfrac{area (ABC)}{area (ADE)}$$
$$\cfrac{area (ABC)}{area (ADE)}=\cfrac{9}{16}$$
Question 2
1 / -0

Which of the following can be used to prove $$\Delta {XYZ}$$ and $$\Delta {UVW}$$ are similar?

A
SSS postulate

B
SAS postulate

C
AAA postulate

D
SSA postulate

Solution

From the figure, $$\Delta {XYZ}$$ and $$\Delta {UVW}$$ are similar by SSS postulate because three sides of one triangle are congruent to three sides of another triangle.
Therefore, the given triangles are similar by SSS similarity postulate.
Question 3
1 / -0

In $$\Delta KLP$$, find (a+b).

A
13 cm

B
9 cm

C
11 cm

D
15 cm

Solution

$$\Delta LST$$ and $$LKP$$
angle L is common
$$\angle LTS=\angle LKP=p^{o}$$
$$\therefore \angle LPK=\angle LST$$
$$\Rightarrow \Delta LST\sim LKP$$ BT AAA
$$\therefore \dfrac{LK}{LT}=\dfrac{LP}{LS}=\dfrac{KP}{ST}$$

$$\Rightarrow \dfrac{8+b}{10}=\dfrac{12}{8}=\dfrac{9}{a}$$

$$\therefore \dfrac{8+b}{10}=\dfrac{12}{8}$$

$$\Rightarrow 64+8b=120$$
$$\Rightarrow b=7$$ cm
$$\therefore \dfrac{12}{8}=\dfrac{9}{a}$$
$$\Rightarrow 12a=72$$
$$\Rightarrow a=6$$ cm
$$\therefore a+b=7+6=13$$ cm
Question 4
1 / -0

Which postulate can be used to prove the triangles are similar?

A
AAA

B
SAS

C
AAS

D
SSS

Solution

From the figure, triangle KLM and triangle RST are similar by SSS postulate because three sides of one triangle are congruent to three sides of another triangle.
Therefore, the given triangles are similar by SSS similarity postulate.
Question 5
1 / -0

Identify the diagrams, based on which the given pair of triangle can be said SSS congruent?

A
figure A

B
figure C

C
figure D

D
figure B

Solution

From the figure C, triangles are similar by SSS condition of congruence.
SSS condition for congruent: Two triangles are said to be congruent if three sides of one triangle are respectively equal to the three sides of the other triangle.
Question 6
1 / -0

When we construct a triangle similar to a given triangle as per given scale factor, we construct on the basis of ...........

A
SSS Similarity

B
AAA similarity

C
Basic proportionality theorem

D
A and C are correct

Solution

As we consider only sides, therefore, SSS similarity is used.
Option A is correct.
Question 7
1 / -0

If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar by which similarity

A
AAA test

B
SSS test

C
SAS test

D
ASA test

Solution

If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the including angles are congruent then the triangles are similar by SAS similarity.
Question 8
1 / -0

$$\triangle ABC$$ is similar to $$\triangle XYZ$$ by $$SAS$$ similarity.If in $$\triangle ABC$$ $$AB=12,BC=8,\angle B=60$$
and in $$\triangle XYZ$$ Find the value of $$\angle Y$$

A
$$240°$$

B
$$60°$$

C
$$15°$$

D
Cannot be determined

Solution

There are two possibilities for the similar triangles according to SAS rule
$$(1) \dfrac{\angle B}{\angle Y}=\dfrac{12}{3}=\dfrac{8}{2}=4\\ \Rightarrow \angle Y=60^\circ$$

$$(2) \angle Y =\angle B=60^\circ$$
Question 9
1 / -0

$$\triangle ABC$$ is similar to $$\triangle XYZ$$ by SSS similarity. If in $$\triangle ABC$$, $$AB=12,BC=8,AC=6$$
and in $$\triangle XYZ,$$ $$XY=6,YZ=4$$. Find the value of $$XZ$$

A
$$2$$

B
$$6$$

C
$$12$$

D
$$3$$

Solution

According to SSS rule,
$$\dfrac{AB}{XY}=\dfrac{BC}{YZ}=\dfrac{CA}{ZX}\\\Rightarrow \dfrac{12}{6}=\dfrac{8}{4}=\dfrac{6}{XZ}\\ \Rightarrow XZ=\dfrac{6}{2}\\\ \ \ \ \ \ \ \ \ \ \ =3$$
Question 10
1 / -0

$$\triangle ABC$$ is similar to $$\triangle XYZ$$ by $$SAS$$ similarity. If in $$\triangle ABC$$ $$AB=12,BC=8,\angle B=60^o$$
and in $$\triangle XYZ$$ $$XY=3$$,$$\angle Y=60^o$$. Find the value of $$YZ$$

A
$$15$$

B
$$20$$

C
$$4$$

D
$$2$$

Solution

If two triangles are similar, then their corresponding sides are proportional.
Given $$\triangle ABC \sim \triangle XYZ$$
$$\implies \dfrac{AB}{XY}=\dfrac{BC}{YZ}$$
$$\implies \dfrac{12}{3}=\dfrac{8}{YZ}$$
$$\implies YZ =\dfrac{3\times 8}{12}$$
$$\implies YZ=2$$