Question

Identify the similar triangles and find x. Then find the measures of the indicated sides.

Answer 1

Answer:

The similar triangles are Δ KMJ and Δ NML

The value of x is 3

KM = 6 and NM = 3

Step-by-step explanation:

* Lets 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  

- Example : In ΔABC and ΔDEF, m∠A = m∠D, m∠B = m∠E and  

 m∠C= m∠F then ΔABC ≈ ΔDEF by AAA  

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.

- Example : In ΔPQR and ΔDEF, m∠P = m∠D, m∠R = m∠F then  

  ΔPQR ≈ ΔDEF by AA  

3) SSS similarity : If the corresponding sides of two triangles are

   proportional, then the two triangles are similar.

- Example : In ΔXYZ and ΔLMN, if  

  then the two triangles are similar by SSS  

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.

- Example : In triangle ABC and DEF, if m∠A = m∠D and  

  then the two triangles are similar by SAS

* Now lets solve the problem

- ∠KMJ is a aright angle and M is on JL

∴ m∠JML = 180° ⇒ straight angle

∵ m∠JMK + m∠LMN = m∠JML

∴ 90° + m∠NML = 180° ⇒ subtract 90° from both sides

∴ m∠NML = 90°

- In Δ KMJ and ΔNML

∵ m∠KMJ = m∠NML ⇒ proved

∵ m∠KJM = m∠NLM ⇒ given

- By using the second case above (AA similarity)

∴ Δ KMJ ≈ Δ NML

* The similar triangles are Δ KMJ and Δ NML

- From similarity

∴ Their sides are proportion

∴ $\frac{KM}{NM}=\frac{MJ}{ML}=\frac{KJ}{NL}$

∵ KJ = 10 and NL = 5

∵ KM = 3 + x and NM = x

- Substitute these values in the proportion relation

∵ $\frac{KM}{NM}=\frac{KJ}{NL}$

∴ $\frac{3+x}{x}=\frac{10}{5}$

- By using cross multiplication

∴ 5(3 + x) = 10(x) ⇒ simplify

∴ 5(3) + 5(x) = 10x

∴ 15 + 5x = 10x ⇒ subtract 5x from both sides

∴ 15 = 5x ⇒ divide both sides by 5

∴ 3 = x

* The value of x is 3

∵ KM = 3 + x

∵ x = 3

∴ KM = 3 + 3 = 6

∵ NM = x

∴ NM = 3

* KM = 6 and NM = 3

- Check the ratio

∵ KM/NM = 6/3 = 2

∵ KJ/NL = 10/5 = 2

∴ The sides are proportion

Answer 2

Answer:

Triangle JMK is similar to triangle LMN.

$x = 3$.

$\rm \overline{KM}= 6$.

$\rm \overline{NM} = 3$.

Step-by-step explanation:

The angle $\rm N\hat{M}L$ is a right angle for it is complementary with another right angle, $\rm K\hat{M}J$.

The diagram also indicates that angle $\rm \hat{J}$ is equal to angle $\rm \hat{L}$. As a result, $\rm \triangle JMK \sim \triangle LMN$ for two of their angles are equal.

Consequently,

$\displaystyle \rm \frac{(\overline{MN})}{(\overline{MK})} = \frac{(\overline{LN})}{(\overline{JK})}$.

Let $x$ be the length of segment $\rm MN$.

$\displaystyle \frac{x}{3+x} = \frac{5}{10}$.

Cross multiply. In other words, multiply both sides by $10(3 + x)$.

$10x = 5(3 + x)$.

$x = 3$.

$\rm \overline{KM} = 3 + \mathnormal{x} = 6$.

$\rm \overline{MN} = \mathnormal{x} = 3$.