Question

In the diagram, StartFraction S Q Over O M EndFraction = StartFraction S R Over O N EndFraction = 4.

Triangles M N O and S R Q are shown. The length of side N O is 8 and the length of side R Q is 48. The length of side O M is 15 and the length of Q S is 60. The length of side M N is 12 and the length of S R is 32.

To prove that the triangles are similar by the SSS similarity theorem, which other sides or angles should be used?

MN and SR
MN and QR
∠S ≅ ∠N
∠S ≅ ∠O

Answer 1

Answer:

MN and QR should be used to prove that the triangles are similar by the SSS similarity theorem ⇒ 2nd answer

Step-by-step explanation:

Let us revise the cases of similarity:

1. AAA similarity : two triangles are similar if all three angles in the first  triangle equal the corresponding angle in the second triangle
2. AA similarity : If two angles of one triangle are equal to the corresponding angles of the other triangle, then the two triangles  are similar.  
3. SSS similarity : If the corresponding sides of the two triangles are  proportional, then the two triangles are similar.  
4. SAS similarity : In two triangles, if two sets of corresponding sides  are proportional and the included angles are equal then the two  triangles are similar.  

In Δs MNO and QRS

∵ SQ = 60

∵ OM = 15

- Find the ratio between SQ and OM

∴ $\frac{SQ}{OM}=\frac{60}{15}=4$

∵ SR = 32

∵ ON = 8

- Find the ratio of SR and ON

∴  $\frac{SR}{ON}=\frac{32}{8}=4$

∵ MN = 12 units

∵ QR = 48

- Find the ratio between QR and MN

∴ $\frac{QR}{MN}=\frac{48}{12}=4$

- The ratio between the corresponding sides are equal

∴ $\frac{SQ}{OM}=\frac{SR}{ON}=\frac{QR}{MN}= 4$

- By using the third case of similarity SSS

∴ Δ MNO is similar to Δ QRS by SSS similarity theorem

∴ You should use sides MN and QR

Answer 2

The given triangles are similar by Side-Side-Side (SSS), similarity when the ratio of the three corresponding pair of sides are equal.

The other sides that should be used to prove that ΔMNO and ΔSRQ are similar are;

• MN and QR

Reasons:

The given parameters are;

$\displaystyle \frac{SQ}{OM} = \mathbf{ \frac{SR}{ON} }= 4$

The triangles are; ΔMNO and ΔSRQ

NO = 8

RQ = 48

OM = 15

QS = 60

MN = 12

SR = 32

Required:

The additional sides to prove that ΔMNO and ΔSRQ are similar by SSS similarity postulate.

Solution:

Based on the given information, we have;

$\displaystyle \frac{SQ}{OM} = \frac{SR}{ON} = 4$

Where ΔMNO and ΔSRQ are similar, the ratio of the corresponding sides are equal, therefore, given that we have;

SQ and SR from ΔSRQ and OM and ON from ΔMNO, the other information on ΔSRQ and ΔMNO are;

• QR in ΔSRQ and MN in ΔMNO

For ΔSRQ and ΔMNO to be similar, the ratio of the remaining sides also gives a value of 4, therefore;

$\displaystyle \mathbf{\frac{QR}{MN} }= 4$

Which gives;

$\displaystyle \frac{SQ}{OM} = \frac{SR}{ON} = \mathbf{ \displaystyle \frac{QR}{MN} }= 4$

Therefore;

To prove that the triangles are similar by SSS, Side-Side-Side, similarity theorem, the sides that should be used are;

• MN and QR

Learn more about SSS similarity postulate here:

https://brainly.com/question/4346515