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Similar Triangles

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 Sub Topics A triangle is said to be similar if they have same shape, but it can have different sizes or size may be similar. If a similar triangle is rotated then also it is called as similar or we can also say that one is mirror image of the other. To understand the similar triangle you have to understand the two main concept one is related shape and the other is related to scale. The shape of a triangle is identified with help of its angles, so we can also say that two Triangles having same shape will also have same Corresponding Angles that preserves their measures. In mathematically term, we can say that ∆ PQR and ∆XYZ are said to be similar triangle if either of the condition gets true: 1. If corresponding sides of the triangle have same ratios of lengthi.e. $\frac{PQ}{ XY}$ = $\frac{QR}{YZ}$ = $\frac{RP}{ZX}$. With this relation we can say that first triangle is an enlargement of the second. 2. If ∠ QPR is equal to ∠ YXZ and ∠ PQR is equal to ∠ XYZ, with this we can say that ∠ PRQ is also equal to ∠ XZY. If ∆ PQR and ∆XYZ are similar then we can write it as ∆ PQR ∼∆XYZ. To represent similar triangles we use either ‘∼’ or we can use three Parallel Lines and both the symbol means “is similar to”. When we have to prove that two triangles are similar then we can use three properties. The very first property says that if triangles have same shape then it is similar and it is known as ‘AA’ property. The next property is called as ‘SSS’ and it reveals that if both triangles have same scale then they are similar. The last property says is ‘SAS’ and it combines the information of both the above mention properties.

Proportional Parts of Triangles

A figure which is a packed or closed figure which consists of three lines and each side is linked end to end and known as triangle. We can say a triangle is a type of a Polygon. There are different types of properties which are:
Triangles have vertex, base, altitude, median, area, perimeter, interior angles, and exterior angles. Now we will see small introduction about all the properties of a triangle:
In a triangle three vertices are present and all the vertices of a triangle depend on the number of sides in a triangle. The opposite side of vertex is known as the base of a triangle. Three medians are present is triangle. The Area of Triangle is half of the base and height. The perimeter of a triangle is the total sum of all the sides of a triangle. Three interior angles are present in a triangle. This is all about the properties of a triangle.
Now we see Proportional Parts of Triangles.
For finding the Proportional Parts of Triangles we have to assume a triangle.

In this given diagram the triangle PQR has the parallel line ST which intersects the other two lines ‘S’ and ‘T’. In this given diagram we have to prove the triangle PQR is equal to the triangle STQ with the help of angle – angle similarity theorem.
So with the help of AA we can write as:
PQ = QR,
SQ QT
Now we use denominator subtraction property for further solving:
So we can write it as:
PQ –QS = QR – QT,
SQ QT
On further solving we get:
When we subtract PQ – QS we get PS and we subtract QR – QT we get TR.
On further solving we get the proportion parts of triangle:
PQ = QR,
SQ QT
By using this method we can find out the proportion parts of triangle.

Similar Triangles Area

If side Ratio and area ratio of two Triangles are equal, then given triangles are called as a similar triangle. Now question arises how we calculate Similar Triangles area:
For evaluation of areas of similar triangles, we use following steps:
Step 1: First we evaluate area of both triangles, which are given in question, by using following formula:
Area of triangle = 1 / 2 * height * width,
Like we have two triangles: triangle ABC and triangle XYZ, where height of triangle ABC is equal to 4 inch and width of triangle ABC is equal to 8, then
Area of triangle ABC = 1 / 2 * height * width,
= 1 / 2 * 4 * 8,
= 2 * 8,
= 16 inch2.
So, area of triangle ABC is equal to 16 inch2.
Similarly, we calculate area of second triangle XYZ, whose height is equal to 12 inch and width is equal to 16 inch, then
Area of triangle XYZ = 1 / 2 * height * width,
= 1 / 2 * 12 * 16,
= 6 * 16,
= 96 inch2
So, area of triangle XYZ is equal to 96 inch2.
Step 2: After evaluation of area of both triangle, now we calculate ratio of both area:
Area of triangle ABC / area of triangle XYZ,
= > 16 / 96,
= > 1 / 6,
So, ratio of two triangles: triangle ABC and triangle XYZ is equal to 1/6.
Step 3: Now, for checking that the given triangles are similar triangles or not, we calculate ratio of their sides:
Ratio of triangle ABC’s side = 4 / 8,
= 1 / 2,
And Ratio of triangle XYZ’s side = 12/ 16,
= 3 / 4.
If ratio of both triangle’s side ratio is equal to ratio of their area, then given triangles are called as a similar triangles:
Ratio of triangle ABC’s side / ratio of triangle XYZ’s side,
= > (1 / 2) / (3 / 4),
= > 1 / 6,
Which is equal to ratio of area? So, given triangles are similar triangles.

Proportional Parts of Similar Triangles

When we talk about the word congruent and similar, the two terms are quite different. When we say we have two Triangles which are similar, it means that the corresponding sides of the two triangles are in proportion. Let us assume two triangles say Δ AB C and Δ P Q R. further let us say that the above given two triangles are similar to each other it means we can write Δ A B C ∞ Δ P Q R from above it is clear that the two triangles are not congruent, but they are only similar to each other. We can say that
∟A ↔ ∟ P,
∟ B↔ ∟Q,
∟ C ↔ ∟ R,
Here we will study about Proportional Parts of Similar Triangles. We say if the two triangles are similar we also say that the Ratio of the corresponding sides of the triangles Δ A B C and Δ P Q R are in proportion. Thus we can write
AB/PQ = BC/QR =CA/RP,
So we conclude that the ratio of Corresponding Parts of the similar triangles always gives the same value for all the three sides.
When we talk about the two similar Δ’s it means that the two triangles looks same but the measure of the line segment of the two triangles is not same. We also observe that two Congruent Triangles are also similar but two similar triangle s are not congruent. We say that even in case of two triangles to be congruent, we say that the ratio of their corresponding sides comes to the equal. Also we say that as in congruent triangles, we have the measure of corresponding sides as exactly same. So we find that their ratio is 1 in all this three sides when Δ’s are congruent but it is not same it figures are similar.

Similar Triangles Perimeter

If Ratio of both triangle’s side ratio is equal to ratio of their perimeter’s, then given Triangles are called as a Similar Triangles by their perimeters. Now we discuss all steps, which we use for evaluation of the similar triangles perimeter:
Step 1: First we calculate perimeter of both triangles, which are given in question, by using following formula:
Perimeter of triangle = side a + side b + side c,
Like we have two triangles: triangle PQR and triangle STU, where all sides of triangle PQR are equal to 3 inch, 4 inch and 5 inch, then
Perimeter of triangle PQR = side p + side q + side r,
= 3 + 4 + 5,
= 12 inch,
So, perimeter of triangle PQR is equal to 12 inch.
Similarly, we can calculate perimeter of second triangle STU, whose all sides are equal to 6 inch, 8 inch and 10 inch, then
Perimeter of triangle STU = side s + side t + side u,
= 6 + 8 + 10,
= 24 inch,
So, perimeter of triangle STU is equal to 24 inch.
Step 2: After evaluation of perimeter of both triangle, now we calculate ratio of both perimeter:
Perimeter of triangle PQR / perimeter of triangle STU
= > 12 / 24,
= > 1 / 2,
So, ratio of both triangles perimeter is equal to 1/2.
Step 3: Now, for checking that the given triangles are similar triangles or not, we calculate ratio between their sides:
Length of side PQ = √ 32 + 42 = √ (9 + 16) = 5,
Length of side PQ = √ 62 + 82 = √ (36 + 64) = 10,

And Ratio of triangles side length = 5/10 = 1 / 2,
Which is equal to ratio of perimeter? So, given triangles are similar triangles.