Created by Turbolearn AI
Similar Shapes Explained
Similarity means that two shapes look the same but may differ in size. A simple
example is zooming in on a picture on a mobile phone; the picture remains the same,
but its size changes.
Identifying Similar Shapes
To determine if shapes are similar, we verify that their angles are the same, even if
the lengths differ.
Example 1: Triangles
In the first pair of triangles, we need to check if their corresponding angles are equal.
Triangle 1 angles: 90°, 30°, 60°
Triangle 2 angles: 90°, 60°, 30°
Since the corresponding angles are equal, the shapes are similar.
Ratio of Sides
For shapes where angle comparison isn't straightforward, we examine if the ratio of
all sides is equal. Similar shapes have sides that increase or decrease proportionally.
Example 2: More Triangles
Consider two triangles where all sides have proportional lengths. If one triangle's
sides are twice the length of the corresponding sides of another triangle, their ratios
are equal.
Triangle 1 sides: 2, 2, 2
Triangle 2 sides: 3, 3, 3
To verify, divide corresponding sides:
2 2 2
= = ≈ 0.666
3 3 3
Page 1
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Since all side ratios are the same approximately0.666, the triangles are similar.
Equilateral Triangles: If all sides of a triangle are equal, it's an equilateral triangle,
and all its angles are 60°. Equilateral triangles are always similar because their
angles are equal.
Non-Similar Shapes
Example 3
If shapes have sides that are equal, but their angles are not equal, then the shapes
are not similar.
Congruent Shapes
Shapes are congruent if two sides and the angle between them are equal.
Example 4
Let's consider two triangles:
Triangle 1: Sides of 6 and 4, with an angle of 120° between them.
Triangle 2: Sides of 4.5 and 3, with an angle of 120° between them.
To check for similarity, calculate the ratios of corresponding sides:
6
= 1.5
4
4.5
= 1.5
3
Since the ratios are equal, the triangles are similar.
Determining Similarity Through Ratios
Example 5
Page 2
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Find the ratios of the corresponding sides
Shape 1 sides: 2, 3, 4
Shape 2 sides: 3, 4, 6
Calculate the ratios:
2
≈ 0.66
3
3
= 0.75
4
4
≈ 0.66
6
Since the ratios are not equal, the shapes are not similar. All sides must have the
same ratio for the shapes to be similar.
Rectangles
Example 6
For rectangles, check if their corresponding angles are equal and if the ratios of their
sides are the same.
Rectangle 1: Sides of 12 and 16
Rectangle 2: Sides of 6 and 8
Ratios:
12
= 2
6
16
= 2
8
Since the ratios are the same and the angles are equal, the rectangles are similar.
Example 7
Page 3
Similar Shapes Explained
Similarity means that two shapes look the same but may differ in size. A simple
example is zooming in on a picture on a mobile phone; the picture remains the same,
but its size changes.
Identifying Similar Shapes
To determine if shapes are similar, we verify that their angles are the same, even if
the lengths differ.
Example 1: Triangles
In the first pair of triangles, we need to check if their corresponding angles are equal.
Triangle 1 angles: 90°, 30°, 60°
Triangle 2 angles: 90°, 60°, 30°
Since the corresponding angles are equal, the shapes are similar.
Ratio of Sides
For shapes where angle comparison isn't straightforward, we examine if the ratio of
all sides is equal. Similar shapes have sides that increase or decrease proportionally.
Example 2: More Triangles
Consider two triangles where all sides have proportional lengths. If one triangle's
sides are twice the length of the corresponding sides of another triangle, their ratios
are equal.
Triangle 1 sides: 2, 2, 2
Triangle 2 sides: 3, 3, 3
To verify, divide corresponding sides:
2 2 2
= = ≈ 0.666
3 3 3
Page 1
, Created by Turbolearn AI
Since all side ratios are the same approximately0.666, the triangles are similar.
Equilateral Triangles: If all sides of a triangle are equal, it's an equilateral triangle,
and all its angles are 60°. Equilateral triangles are always similar because their
angles are equal.
Non-Similar Shapes
Example 3
If shapes have sides that are equal, but their angles are not equal, then the shapes
are not similar.
Congruent Shapes
Shapes are congruent if two sides and the angle between them are equal.
Example 4
Let's consider two triangles:
Triangle 1: Sides of 6 and 4, with an angle of 120° between them.
Triangle 2: Sides of 4.5 and 3, with an angle of 120° between them.
To check for similarity, calculate the ratios of corresponding sides:
6
= 1.5
4
4.5
= 1.5
3
Since the ratios are equal, the triangles are similar.
Determining Similarity Through Ratios
Example 5
Page 2
, Created by Turbolearn AI
Find the ratios of the corresponding sides
Shape 1 sides: 2, 3, 4
Shape 2 sides: 3, 4, 6
Calculate the ratios:
2
≈ 0.66
3
3
= 0.75
4
4
≈ 0.66
6
Since the ratios are not equal, the shapes are not similar. All sides must have the
same ratio for the shapes to be similar.
Rectangles
Example 6
For rectangles, check if their corresponding angles are equal and if the ratios of their
sides are the same.
Rectangle 1: Sides of 12 and 16
Rectangle 2: Sides of 6 and 8
Ratios:
12
= 2
6
16
= 2
8
Since the ratios are the same and the angles are equal, the rectangles are similar.
Example 7
Page 3
