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Q1(iii):
Fill in the blanks using the correct word given in brackets : (iii) All ______ triangles are similar. (isosceles, equilateral)
Solution :
Given: A statement regarding the similarity of specific types of triangles: "All ______ triangles are similar."
To Find: The correct word from the options (isosceles, equilateral) that completes the statement to make it mathematically true.
Step 1: Understanding the Definition of Similar Triangles
Two triangles are said to be similar if:
1. Their corresponding angles are equal.
2. Their corresponding sides are in the same ratio (proportion).
Step 2: Analyzing Equilateral Triangles
An equilateral triangle is a triangle in which all three sides are equal in length and all three interior angles are equal to $60^\circ$.
Let $T_1$ and $T_2$ be two equilateral triangles with side lengths $s_1$ and $s_2$ respectively.
- The angles of $T_1$ are $60^\circ, 60^\circ, 60^\circ$.
- The angles of $T_2$ are $60^\circ, 60^\circ, 60^\circ$.
Since the corresponding angles are equal ($60^\circ = 60^\circ$), the condition for similarity is satisfied regardless of the side lengths. Thus, all equilateral triangles are similar.
Step 3: Analyzing Isosceles Triangles
An isosceles triangle is a triangle with at least two equal sides. Consider two isosceles triangles:
- Triangle 1: Sides $5, 5, 8$. The angles are approximately $36.87^\circ, 71.56^\circ, 71.56^\circ$.
- Triangle 2: Sides $5, 5, 2$. The angles are approximately $151.04^\circ, 14.48^\circ, 14.48^\circ$.
Since the corresponding angles are not equal, isosceles triangles are not necessarily similar.
Step 4: Conclusion
Based on the geometric properties established in Step 2, the condition of similarity holds true for all equilateral triangles because their angular configuration is constant ($60^\circ$ each).
Final Answer: All equilateral triangles are similar.
More Questions from Class 10 Mathematics Triangles EXERCISE 6.1
- Q1(i): Fill in the blanks using the correct word given in brackets : (i) All circles are ______. (congruent, similar)
- Q1(ii): Fill in the blanks using the correct word given in brackets : (ii) All squares are ______. (similar, congruent)
- Q1(iv): Fill in the blanks using the correct word given in brackets : (iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ______ and (b) their corresponding sides are ______. (equal, proportional)
- Q2(i): Give two different examples of pair of (i) similar figures.
- Q2(ii): Give two different examples of pair of (ii) non-similar figures.
- Q3: State whether the following quadrilaterals are similar or not:
CBSE Solutions for Class 10 Mathematics Triangles
Chapters in CBSE - Class 10 Mathematics
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