Refer to the picture above. Are these two triangles similar? If so, What is the theorem that allows you to state that these triangles are similar?
A)Yes they are similar Side-Angle-Side (SAS) Similarity Theorem
B)Yes they are similar by Side-Side-Side (SSS) Similarity Theorem
C)These triangles are not similar.

2) What is the length of segment LP?
A) 13.75
B)35.2
C)23.2

Question 1

We first determine all pairs of side that correspond

Side GE correspond with side IJ
Side EF correspond with side KI
Side FG correspond with side JK

Then we see if all pairs have the same ratio of their length

GE : IJ = 90 : 120 = 3 : 4
EF : IK = 117 : 156 = 3 : 4
FG : JK = 135 : 180 = 3 : 4

Since all sides have the same ratio, hence triangle GEF is similar to the triangle JIK by properties SSS

Question 2

We start by listing all the corresponding sides

Side LO correspond with side LM
Side LP correspond with side LN
Side OP correspond with NM

We can then write down each pair in term of the ratio of their side length
LO : LM = 22 : 30 = 11 : 15
LP : LN = [tex](x+12):(x+12+5)[/tex] = [tex](x+12):(x+17)[/tex]

Although we don't know the measurement of side OP and NM, we have enough information to solve the problem

We know from the ratio that side LM is [tex] \frac{15}{11} [/tex] times side LO
so side LN is also [tex] \frac{15}{11} [/tex] times side LP

[tex]LN= \frac{15}{11}LP [/tex]
[tex]x+17=( \frac{15}{11})(x+12) [/tex]
[tex]x+17= \frac{15x+180}{11} [/tex]
[tex]11(x+17)=15x+180[/tex]
[tex]11x+187=15x+180[/tex]
[tex]187-180=15x-11x[/tex]
[tex]15x-11x=187-180[/tex]
[tex]4x=7[/tex]
[tex]x=1.75[/tex]

Length of line segment LP is 1.75+12=13.75 (option A)

AkshayG AkshayG · Answer 2 · 2019-11-30T09:59:13+01:00

Part (a)- The triangles are similar by Side-Side-Side (SSS) Similarity Theorem.

Part (b)- The length of LP is [tex]\boxed{13.75}.[/tex]

Further Explanation:

The similar triangles are those in which all the corresponding angles are equal and the sides are proportional.

There are many similarity rules and are as follows.

1. Angle Angle (AA)

2. Side SideSide (SSS)

3. Side Angle Side (SAS)

Given:

Part (a)

The options are as follows,

A) Yes they are similar Side-Angle-Side (SAS) Similarity Theorem.

B) Yes they are similar Side-Angle-Side (SSS) Similarity Theorem.

C) These triangles are not similar.

Part (b)

The options are as follows,

A) 13.75

B) 35.2

C) 23.2

Explanation:

Part (a)

In [tex]{\Delta{\text {EFG} \:{\text{and}\: \Delta{\text{ IJK}.[/tex]

[tex]\begin{aligned}\frac{{{\text{EG}}}}{{{\text{IJ}}}}&= \frac{{90}}{{120}}\\&= \frac{3}{4}\\\end{aligned}[/tex]

[tex]\begin{aligned}\frac{{{\text{GF}}}}{{{\text{JK}}}}&= \frac{{135}}{{180}}\\&= \frac{3}{4}\\\end{aligned}[/tex]

[tex]\begin{aligned}\frac{{{\text{EF}}}}{{{\text{IK}}}}&= \frac{{117}}{{156}}\\&= \frac{3}{4}\\\end{aligned}[/tex]

The ratios of corresponding sides are equal. Therefore, triangle EGF is similar to triangle IJK by SSS similarity theorem.

Option B is correct.

Part (b)

The [tex]\Delta {\text{MLN and }}\Delta {\text{OLP}}[/tex] are similar to each other. Therefore, the ratios of the corresponding sides are equal.

[tex]\begin{aligned}\frac{{{\text{ML}}}}{{{\text{OL}}}} &= \frac{{{\text{LN}}}}{{{\text{LP}}}}\\\frac{{30}}{{22}} &= \frac{{x + 12 + 5}}{{x + 12}}\\\frac{{15}}{{11}} &= \frac{{x + 17}}{{x + 12}}\\11\times \left( {x + 17} \right) &= 15 \times \left( {x + 12} \right)\\11x + 187 &= 15x + 180\\\end{aligned}[/tex]

Further solve the above equation.

[tex]\begin{aligned}11x + 187&= 15x + 180\\187 - 180 &= 15x - 11x\\7 &= 4x\\\frac{7}{4}&=x\\1.75&= x\\\end{aligned}[/tex]

The length of LP can be obtained as follows,

[tex]\begin{aligned}{\text{LP}} &= 12 + 1.75\\&=13.75\\\end{aligned}[/tex]

Option (A) is correct.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Triangle

Keywords: similar, two triangles, theorem, length, segment, LP, congruent, angles, triangle, ASA, angle side angle, congruent sides, acute angle, side, corresponding angles, congruent triangle, similarity theorem, SSS congruency theorem, SSS similarity theorem, AA similarity postulate, SAS congruency postulate.