**Finding the Missing Side Length with the Concept Similar Triangles :**

In this section, we will learn, how to find the missing sides of the triangle using the concept similar triangles.

There are three ways to prove that two triangles are similar :

- Side-Side-Side (SSS) Similarity Theorem
- Side-Angle-Side (SAS) Similarity Theorem
- Angle-Angle (AA) Similarity Postulate.

To know more about proving similar triangles, please visit the page "Proving Triangles are Similar".

**Example 1 :**

Find the unknown values in each of the following figures. All lengths are given in centimeters (all measures are not in scale)

**Solution :**

From triangles ABC and triangle ADE

∠ABC = ∠ADE (corresponding angles)

∠A = ∠A (common angle)

So ∆ ABC ~ ∆ ADE

(AC/AE) = (BC/DE)

[x/(x + 8)] = (8/24)

[x/(x + 8)] = (1/3)

3 x = 1 (x + 8)

3 x = x + 8

3 x – x = 8

2 x = 8

x = 8/2

x = 4 cm.

also ∆ EAG and ∆ ECF are congruent triangles

So, (EC/EA) = (CF/AG)

AG = (EA ⋅ CF)/EC

EA = EC + CA

= 8 + 4 ==> 12 cm

y = (6 ⋅ 12)/8

y = 72/8

= 9 cm

The values of x and y are 4 and 9 cm respectively.

(ii) In ∆ HBC, the sides FG and BC are parallel

∆ HFG ~ ∆ HBC

(HF/HB) = (FG/BC)

(4/10) = (x/9)

x = (4 ⋅ 10)/4

x = 3.6 cm

In triangle ∆ FBD and ∆FHG the sides BD and GH are parallel,

∠FBD = ∠FHG (alternate angles)

∠BFD = ∠HFG (vertically opposite angels)

By using AA similarity criterion ∆ FBD ~ ∆ FHG

(FG/FD) = (FH/FB)

[x/(y + 3)] = (4/6)

3.6/(y + 3) = (2/3)

3.6(3) = 2 (y + 3)

10.8 = 2 y + 6

2 y = 10.8 – 6

2 y = 4.8

y = 4..8/2

y = 2.4 cm

In triangles ∆ AEG and ∆ ABC, the sides EG and BC are parallel,

∠A = ∠A (common angle)

∠AEG = ∠ABC (corresponding angles)

By using AA similarity criterion ∆ AEG ~ ∆ ABC

(AE/AB) = (EG/BC)

[z/(z + 5 )] = (x + y)/9

[z/(z + 5 )] = 6/9

9 z = 6 (z + 5)

9 z = 6 z + 30

9 z – 6 z = 30

3 z = 30

z = 30/3

z = 10 cm

So length of the side FG = 3.6 cm

Length of the side BF = 2.4 cm

Length of the side AE = 10 cm

After having gone through the stuff given above, we hope that the students would have understood, finding the missing side length with the concept similar triangles.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**