Question

traingle ABC and triangle PQR are equilateral triangles.If A(triangle ABC): A(triangle PQR)=1:16 and AB=2cm then what is the length of PR? ​
choose the correct alternative options.
1)4cm 2)2cm 3)6cm 4)8cm

Answer 1

Answer:

Since both are equilateral triangles, they both are similar triangles.

Hence, their side ratios will also be equal.

Hence, AB/PQ = 1/16

If AB is 2cm, then PQ is 8cm

Since they are equilateral triangles, PR = PQ = QR = 8cm

Answer 2

Answer:

The length of PR is 8 cm.

Option ( 4 ) 8 cm

Step-by-step-explanation:

We have given that,

$\sf\:\triangle\:ABC\:\&\:\triangle\:PQR\:are\:equilateral\:triangles\\\\\\\therefore\sf\:\angle\:A\:=\:\angle\:B\:=\:\angle\:C\:=\:60^{\circ}\\\\\\\sf\:\angle\:P\:=\:\angle\:Q\:=\:\angle\:R\:=\:60^{\circ}\\\\\\\therefore\:\triangle\:ABC\:\sim\:\triangle\:PQR\:\:\:-\:-\:[\:A\:-\:A\:-\:A\:test\:]\\\\\\\sf\:Now,\\\\\\\pink{\sf\:\dfrac{A\:(\:\triangle\:ABC\:)}{A\:(\:\triangle\:PQR\:)}\:=\:\dfrac{AB^2}{PR^2}}\:\:\:-\:-\:[\:Ratio\:of\:areas\:of\:two\:triangles\:]\\\\\\\implies\sf\:\dfrac{1}{16}\:=\:\dfrac{\:(\:2\:)^2\:}{PR^2}\:\:\:-\:-\:[\:Given\:]\\\\\\\implies\sf\:\dfrac{1}{16}\:=\:\dfrac{4}{PR^2}\\\\\\\implies\sf\:PR^2\:=\:4\:\times\:16\\\\\\\implies\sf\:PR\:=\:\sqrt{4\:\times\:16}\:\:-\:-\:[\:Taking\:square\:roots\:]\\\\\\\implies\sf\:PR\:=\:2\:\times\:4\\\\\\\implies\boxed{\red{\sf\:PR\:=\:8\:cm}}$

Additional Information:

1. Similar triangles:

Two triangles are similar if their corresponding angles are congruent and the corresponding side are in proportion.

2. Area of two similar triangles:

If two triangles are similar, then the ratio of areas of those triangles is equal to the ratio of squares of the corresponding sides.

3. Tests of similarity of triangles:

A. AAA test

B. AA test

C. SAS test

D. SSS test

4. AAA test:

In two triangles, if the all corresponding angles are congruent, then the two triangles are similar.

5. AA test:

In two triangles, if only two corresponding angles are congruent, then the two triangles are similar.

6. SAS test:

In two triangles, if two corresponding sides are in proportion and the corresponding angle made by them is congruent, then two triangles are similar.

7. SSS test:

In two triangles, if all three corresponding sides are in proportion, then the two triangles are similar.