Table of contents | |
Introduction | |
Similar Figures | |
Similarity of Triangles | |
Criteria For Similarity of Triangles | |
Summary |
Two triangles are said to be similar triangles if:
For example:
If in ∆ABC and ∆PQR
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and,
AB/PQ = BC/QR = AC/PR
Then, △ABC∼△PQR
where the symbol ∼ is read as ‘is similar to’.
Conversely
If △ABC is similar to △PQR, then
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and,
AB/PQ = BC/QR = AC/PRNote: The ratio of any two corresponding sides in two equiangular triangles is always the same.
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then other two sides are divided in the same ratio.
This theorem is known as the Basic Proportionality Theorem (BPT) or Thales theorem.
Given:
To Proof:
Proof:
Now,
Now,
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Given: In ΔABC, DE is a straight line such that
To prove DE || BC.
Construction: If DE is not parallel to BC, draw DF meeting AC at F.
Proof: In ΔABC, let DF || BC
[∴ A line drawn parallel to one side of a Δ divides the other two sides in the same ratio.]
⇒ FC = EC.
It is possible only when E and F coincide
Hence, DE || BC.
Example 1: If a line intersects sides AB and AC of a triangle ABC at D and E respectively and is parallel to BC,
prove that
Solution:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Theorem 3 : If in two triangles, if corresponding angles are equal, then their corresponding sides are in the same ratio i.e., they are proportional, and hence the two triangles are similar.
This criterion is referred to as the AAA (Angle–Angle–Angle) criterion of similarity of two triangles.
If ∠A=∠C and ∠B=∠D then △ABC∼△DEF
Example: In theΔABC length of the sides are given as AP = 5 cm , PB = 10 cm and BC = 20 cm. Also PQ||BC. Find PQ.
Solution: In ΔABC and ΔAPQ, ∠PAQ is common and ∠APQ = ∠ABC (corresponding angles)
⇒ ΔABC ~ ΔAPQ (AA criterion for similar triangles)
⇒ AP/AB = PQ/BC
⇒ 5/15 = PQ/20
⇒ PQ = 20/3 cm
If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similiar
This criterion is referred to as the SSS (Side–Side–Side) similarity criterion for two triangles.
Using Theorem: if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. This theorem is often referred to as the Basic Proportionality Theorem or Thales' Theorem.
Example: Two triangles ABC and DEF are similar such that AB = 8cm, BC = 10cm, CA =y cm, DE = 6 cm, EF = x cm and FD = 9 cm . Find the Values of x and y?
Solution: As △ABC∼△DEF then AB/DE=AC/DF=BC/EF
So now putting values 8/6=y/9=10/x
8/6=y/9 and 8/6=10/x
4/3=y/9 and 4/3=10/x
4*9=y*3 and 4*x=10*3 (Cross multiplying)
so y comes out to be =12
and x=7.5
This criterion is referred to as the SAS (Side–Angle–Side) similarity criterion for two triangles.
If AB/ED=BC/EF and ∠B=∠E Then △ABC∼△DEF
Example: Determine if the following triangles are similar. If so, write the similarity criteria
Solution:
We can see that ∠B≅∠F and these are both included angles. We just have to check that the sides around the angles are proportional.
ABDFBCFE=128=32=2416=32
Since the ratios are the same ΔABC∼ΔDFE by the SAS Similarity Theorem.
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1. What are similar figures? |
2. How can you determine if two triangles are similar? |
3. What are the criteria for similarity of triangles? |
4. Can triangles with different side lengths be similar? |
5. How is similarity of triangles useful in real life applications? |
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