Steps to Draw and Prove Similar Triangles in Geometry
Construction of Similar Triangles Class 10
Here, construction of similar triangles is given as per scale factor. Scale factor refers to the ratio of the sides of the triangle to be drawn with the corresponding sides of the given triangle.
The construction of similar triangle involves two different situations:
(i) The triangle to be drawn is smaller than the given triangle; here scale factor is less than 1.
(ii) The triangle to be drawn is larger than the given triangle, here scale factor is greater than 1.
Understanding Construction of Similar Triangles Using Real-Life Example
To understand the concept of similarity or similarity of triangles, imagine Eiffel Tower. Now imagine in your mind a mini version of it. Did you understand? The mini version is actually just a scaled-down version of the original monument. The shape remains the same just the size alters. Same is the case with the Triangles. This below figure will help you understand better.
[Image will be Uploaded Soon]
Let’s get to know how to check the similarity of triangles using the solve questions.
AA Similarity Theorem to Test the Similarity of Triangles
Angle Angle (AA) similarity theorem implies that two triangles must be similar to one another provided they consist of two corresponding angles in a manner that they are equal or congruent in measurement.
Applying this theorem, there will be no requirement of displaying that all three corresponding angles belonging to two triangles are in congruence for the purpose of proving that they are similar.
Pythagoras Theorem to Test and Prove Similarity of Triangles
In a right angled triangle the square of the longest side i.e. the hypotenuse is equivalent to the sum of squares of the remaining two sides.
Solved Examples
Example1: ABC is a right-angle triangle right angled at B. MN is parallel to BC. AB = 6 cm and AM: MB = 1: 2. Find out the lengths of AC and BC.
Solution:
Given,
AM / MB = ½
MB / AM = 2
AM + MB / AM = 2+1
AB/AM = 3
In △ ABC and △ AMN
∠MAN=∠BAC (Common angle)
∠AMN=∠ABC =90⁰ (MN II BC)
∠MNA=∠BCA (3rd angle)
Thus, △ABC∼△AMN (Application of AAA rule)
Therefore,
AM/AB = MN/BC= AN/AC (Corresponding sides)
AN/AC = 3
Thus, AC = 3 × AC = 3×4 = 12cm
Now, in △ABC,
AB² + BC² = AC² (Applying the Pythagoras theorem)
6² + BC² =12²
BC² = 144−36
BC =√108
BC =10.392cm
Example 2: Draw a triangle similar to a given triangle PQR in a way that each of its sides is (5/7)th of the corresponding sides of ∆PQR. It is given that
PQ = 4 cm, PR = 5 cm and QR = 6 cm.
Solution:
Follow these Steps of Construction in an orderly way:
Step I: Construct a line segment QR = 6 cm.
Step II: With Q as centre point and radius = PQ = 4 cm, make an arc.
Step III : With R as centre point and radius = PR = 5 cm, draw another arc, bisecting the arc drawn in step II at P.
Step IV: Join PQ and PR to get the triangle PQR.
Step V: Below base QR, draw an acute angle ∠RQX.
Step VI: Along QX, mark 7 points Q1, Q2, Q3, Q4, Q5, Q6, Q7 such that
QQ1 = Q1Q2 = …… = Q6Q7.
Step VII: Connect Q7R.
Step VIII: Because we have to construct a triangle each of whose sides is (5/7)th of the corresponding sides of ∆PQR. So take 5 parts out of 7 equal parts on QX i.e. from Q5, Draw Q5R´ || Q7R, bisecting QR at R´.
Step IX: From R´, draw R´P´ || RP, meeting QP at P´.
∆P´QR´ is the needed triangle each of whose sides is (5/7)th of the corresponding sides of ∆PQR.
FAQs on How to Construct Similar Triangles: Methods and Examples
1. What does it mean for two triangles to be similar in Geometry?
Two triangles are considered similar if they satisfy two key conditions: their corresponding angles are equal, and their corresponding sides are in the same ratio or proportion. In simple terms, they have the exact same shape but can be of different sizes.
2. What are the main criteria to check if two triangles are similar as per the CBSE Class 10 syllabus?
To prove that two triangles are similar, you can use any of the following three criteria:
- AAA (Angle-Angle-Angle): If all three corresponding angles of two triangles are equal, the triangles are similar. This is often simplified to just AA, as the third angle will automatically be equal.
- SSS (Side-Side-Side): If the lengths of the corresponding sides of two triangles are in the same proportion, they are similar.
- SAS (Side-Angle-Side): If two corresponding sides of two triangles are in proportion and the angle included between these sides is equal, the triangles are similar.
3. What are the basic steps to construct a triangle similar to a given triangle?
To construct a triangle similar to a given ΔABC with a specific scale factor, you generally follow these steps:
- Draw a ray BX starting from vertex B, making an acute angle with the base BC.
- Mark a number of equally spaced points on the ray BX that corresponds to the larger number in the scale factor's fraction.
- Join the relevant point on the ray to vertex C and then draw a line parallel to it from the other relevant point, intersecting the base BC at a new point, C'.
- From C', draw a line parallel to side AC, which intersects side AB at A'. The new triangle, ΔA'BC', is the required similar triangle.
4. What is the role of the 'scale factor' in constructing similar triangles?
The scale factor is a ratio (e.g., 3/4 or 5/2) that determines the size of the new triangle relative to the original. It is the most critical piece of information for the construction. If the scale factor is less than 1 (like 3/4), the new triangle will be smaller than the original. If the scale factor is greater than 1 (like 5/2), the new triangle will be larger.
5. How does the construction process differ when the scale factor is 2/5 compared to 5/2?
The core principle remains the same, but the connection points and the final position of the triangle change:
- For a scale factor of 2/5 (less than 1): The new triangle will be constructed inside the original triangle. You would connect the 5th point on your construction ray to the original vertex and draw the parallel line from the 2nd point.
- For a scale factor of 5/2 (greater than 1): The new triangle will be constructed outside the original triangle. This requires you to first extend the sides of the original triangle. You would connect the 2nd point on the ray and draw the parallel line from the 5th point.
6. Why does the method of drawing parallel lines work for this construction?
This construction method is a practical application of the Basic Proportionality Theorem (BPT) and its converse. When you draw a line parallel to one side of a triangle, it divides the other two sides proportionally. This action ensures that the sides of the new triangle are in the exact same ratio as the sides of the original triangle, which is the fundamental condition for similarity.
7. What are some real-world applications of constructing similar triangles?
The concept of similar triangles is fundamental in various fields. It is used in:
- Architecture and Engineering: To create scaled-down blueprints and models of buildings and structures.
- Cartography (Map Making): Maps are similar representations of geographical areas, scaled down for practical use.
- Art and Photography: Artists use principles of similarity for perspective and proportion.
8. What are common mistakes students make when constructing a similar triangle for exams?
Some common errors to avoid during the construction of similar triangles include:
- Incorrectly connecting points: For a scale factor m/n, mixing up which point (m or n) to connect to the vertex first.
- Drawing non-parallel lines: The parallel lines must be accurate. It is best to use a compass and ruler or a set square to ensure they are truly parallel.
- Forgetting to extend lines: When the scale factor is greater than 1, students often forget to extend the original sides of the triangle to complete the construction of the larger, similar triangle.
