Two triangles are said to be similar when one can be obtained from the other by uniformly scaling. The ratio of the area of two similar triangles is equal to the square of the ratio of any pair of the corresponding sides of the similar triangles. If two triangles are similar it means that: All corresponding angle pairs are equal and all corresponding sides are proportional. However, in order to be sure that the two triangles are similar, we do not necessarily need to have information about all sides and all angles.
For similar triangles, not only do their angles and sides share a relationship, but also the ratio of their perimeter, altitudes, angle bisectors, areas, and other aspects are in proportion. Let us study and understand the relation between the area of similar triangles in the following sections.
Area of Similar Triangles Theorem
Area of similar triangles theorem help in establishing the relationship between the areas of two similar triangles. It states that "The ratio of the areas of two similar triangles is equal to the square of the ratio of any pair of their corresponding sides". Consider the following figure, which shows two similar triangles, ΔABC and ΔDEF.
According to the theorem for area of similar triangles, Area of ΔABC/Area of ΔDEF = (AB)2/(DE)2 = (BC)2/(EF)2 = (AC)2/(DF)2. We will understand the proof of this theorem in the next section.
Proof of Area of Similar Triangles Theorem
Statement: The ratio of the areas of two similar triangles is equal to the square of the ratio of any pair of their corresponding sides.
Given: Consider two triangles, ΔABC and ΔDEF, such that ΔABC∼ΔDEF
To prove: Area of ΔABC/Area of ΔDEF = (AB)2/(DE)2 = (BC)2/(EF)2 = (AC)2/(DF)2
Construction: Draw the altitudes AP and DQ to the sides BC and EF respectively, as shown below:
Proof: Since, ∠B = ∠E, [ ∵ ΔABC ~ ΔDEF ] and,
∠APB = ∠DQE.....[ ∵ AP and DQ are perpendicular on sides BC and EF respectively ⇒ Both angles are equal to 90º ]
By, AA property of similarity of triangles, we can note that ΔABP and ΔDEQ are equiangular.
Hence, ΔABP ~ ΔDEQ
Thus, AP/DQ = AB/DE
This further implies that,
AP/DQ = BC/EF ----- (1)....[ ∵ ΔABC∼ΔDEF ⇒ AB/DE = BC/EF]
Thus,
Area(ΔABC)/Area(ΔDEF) = [(1/2) × BC × AP]/[(1/2) × EF × DQ] = (BC/EF) × (AP/DQ) = (BC/EF) × (BC/EF) ....[from (1)]
⇒ Area(ΔABC)/Area(ΔDEF) = (BC/EF)2
Similarly, we can show that,
Area of ΔABC/Area of ΔDEF = (AB)2/(DE)2 = (BC)2/(EF)2 = (AC)2/(DF)2
Challenging Question:
It is given that ΔABC ~ ΔXYZ. The area of ΔABC is 45 sq units and the area of ΔXYZ is 80 sq units. YZ = 12 units. Find BC? Hint: Use Theorem for Area of Similar Triangles.
Important Notes on Area of Similar Triangles
- The ratio of the areas of two similar triangles is equal to the square of the ratio of any pair of their corresponding sides.
- For similar triangles ΔABC and ΔDEF, Area of ΔABC/Area of ΔDEF = (AB)2/(DE)2 = (BC)2/(EF)2 = (AC)2/(DF)2
- All corresponding angle pairs are equal and all corresponding sides are proportional for similar triangles.
Related Topics on Area of Similar Triangles
- Similar Triangles
- Similar Triangles Formulas
- What is Similarity?
-
Example 1: Consider two similar triangles, ΔABC and ΔDEF, as shown below:
AP and DQ are medians in the two triangles. Show that
ArΔ(ABC)/AP2 = ArΔ(DEF)/DQ2 using areas of similar triangles theorem.
Solution: Since ΔABC ~ ΔDEF,
AB/DE = BC/EF
⇒AB/DE = (1/2)BC/(1/2)EF
⇒AB/DE = BP/EQ →(1)
Also,
∠B = ∠E ----- (2) ... [ ∵ ΔABC ~ ΔDEF]
From (1) and (2) and by SAS similarity criterion, We can note that,
ΔABP ~ ΔDEQ
⇒AB/DE = AP/DQ →(3)
Now, by theorem for areas of similar triangles,
ArΔ(ABC)/ArΔ(DEF) = AB2/DE2 = AP2/DQ2 ....[from (3)]
⇒ArΔ(ABC)/AP2 = ArΔ(DEF)/DQ2 -
Example 2: Consider the following figure:
It is given that XY || BC and divides the triangle into two parts of equal areas. Find the ratio AX: XB using the area of similar triangles theorem.
Solution: Since XY || BC, ∠X = ∠B and ∠Y = ∠C ...[Corresponding angles]
⇒ΔAXY must be similar to ΔABC...[By AA similarity criterion in triangles]Now, by theorem for area of similar triangles,
Ar(ΔABC)/Ar(ΔAXY) = AB2/AX2 → (1)
Also, XY divides the triangle into two parts of equal areas. Thus,
Ar(ΔABC)/Ar(ΔAXY) = 2 → (2)
From (1) and (2), we have,
AB2/AX2 = 2
⇒AB/AX = √2
⇒(AB/AX) − 1 = √2 − 1
(AB - AX) / (AX) = √2 − 1
⇒XB/AX = √2 − 1
⇒AX/XB = 1/(√2 − 1)
Show Solution >
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FAQs on Area of Similar Triangles
The area of two similar triangles shares a relationship with the ratio of the corresponding sides of the similar triangles. According to the area of similar triangles theorem, we can state that "the ratio of the areas of two similar triangles is equal to the square of the ratio of any pair of their corresponding sides".
What Is the Ratio of Area of Similar Triangles?
The ratio of the area of two similar triangles is equal to the square of the ratio of any pair of the corresponding sides of the similar triangles. For example, for any two similar triangles ΔABC and ΔDEF,
Area of ΔABC/Area of ΔDEF = (AB)2/(DE)2 = (BC)2/(EF)2 = (AC)2(DF)2.
What is the Relation Between Two Similar Triangles Area and Length of the Sides?
The ratio of the areas of two similar triangles is equal to the square of the ratio of any pair of the corresponding sides of the similar triangles.
Do Similar Triangles Have Equal Areas?
Similar triangles will have the ratio of their areas equal to the square of the ratio of their pair of corresponding sides. So, the areas of two triangles cannot be necessarily equal. But note that congruent triangles always have equal areas.
How Do You Solve For Areas of Two Similar Triangles?
Areas of similar triangles can be solved by relating their ratio with the ratio of the pair of corresponding sides. For any two similar triangles, the ratio of the areas is equal to the square of the ratio of corresponding sides.
What Is the Areas of Similar Triangles Theorem?
The areas of similar triangles theorem state that "the ratio of the area of two similar triangles is equal to the square of the ratio of any pair of their corresponding sides"
How to Prove Theorem For Areas of Similar Triangles?
The theorem for the areas of similar triangles can be proved by constructing altitudes for both triangles and comparing the area thus obtained with the ratio of corresponding sides of both the similar triangles. To understand the proof in detail, refer to section Proof of Areas of Similar Triangles Theorem of this page.
Two triangles are similar if they have:
- all their angles equal
- corresponding sides are in the same ratio
But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough.
There are three ways to find if two triangles are similar: AA, SAS and SSS:
AA
AA stands for "angle, angle" and means that the triangles have two of their angles equal.
If two triangles have two of their angles equal, the triangles are similar.
So AA could also be called AAA (because when two angles are equal, all three angles must be equal).
SAS
SAS stands for "side, angle, side" and means that we have two triangles where:
- the ratio between two sides is the same as the ratio between another two sides
- and we we also know the included angles are equal.
If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
In this example we can see that:
- one pair of sides is in the ratio of 21 : 14 = 3 : 2
- another pair of sides is in the ratio of 15 : 10 = 3 : 2
- there is a matching angle of 75° in between them
So there is enough information to tell us that the two triangles are similar.
Using Trigonometry
We could also use Trigonometry to calculate the other two sides using the Law of Cosines:
In Triangle ABC:
- a2 = b2 + c2 - 2bc cos A
- a2 = 212 + 152 - 2 × 21 × 15 × Cos75°
- a2 = 441 + 225 - 630 × 0.2588...
- a2 = 666 - 163.055...
- a2 = 502.944...
- So a = √502.94 = 22.426...
In Triangle XYZ:
- x2 = y2 + z2 - 2yz cos X
- x2 = 142 + 102 - 2 × 14 × 10 × Cos75°
- x2 = 196 + 100 - 280 × 0.2588...
- x2 = 296 - 72.469...
- x2 = 223.530...
- So x = √223.530... = 14.950...
Now let us check the ratio of those two sides:
a : x = 22.426... : 14.950... = 3 : 2
the same ratio as before!
Note: we can also use the Law of Sines to show that the other two angles are equal.
SSS
SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
In this example, the ratios of sides are:
- a : x = 6 : 7.5 = 12 : 15 = 4 : 5
- b : y = 8 : 10 = 4 : 5
- c : z = 4 : 5
These ratios are all equal, so the two triangles are similar.
Using Trigonometry
Using Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:
In Triangle ABC:
- cos A = (b2 + c2 - a2)/2bc
- cos A = (82 + 42 - 62)/(2× 8 × 4)
- cos A = (64 + 16 - 36)/64
- cos A = 44/64
- cos A = 0.6875
- So Angle A = 46.6°
In Triangle XYZ:
- cos X = (y2 + z2 - x2)/2yz
- cos X = (102 + 52 - 7.52)/(2× 10 × 5)
- cos X = (100 + 25 - 56.25)/100
- cos X = 68.75/100
- cos X = 0.6875
- So Angle X = 46.6°
So angles A and X are equal!
Similarly we can show that angles B and Y are equal, and angles C and Z are equal.
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