s Geometry Standard Worksheet with solutions, key axioms, and exam practice questions on SJMaths."> s Geometry Standard Worksheet with solutions, key axioms, and exam practice questions on SJMaths.">
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Chapter 5: Introduction to Circles

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This page provides comprehensive Chapter 5: Introduction to Circles - Standard Worksheet - SJMaths. Standard level practice worksheet for Class 9 Introduction to Circles.

Standard Level Worksheet

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  1. Question 1: If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$, then prove that $AC = \frac{1}{2}AB$.
    Solution: Given $AC = BC$. Add $AC$ to both sides: $AC + AC = BC + AC$.
    $2AC = AB$ (since $BC+AC$ coincides with $AB$).
    $\therefore AC = \frac{1}{2}AB$.
  2. Question 2: Solve the equation $x - 5 = 15$ and state the axiom used.
    Solution: $x - 5 = 15$. Add 5 to both sides: $x - 5 + 5 = 15 + 5 \Rightarrow x = 20$.
    Axiom: "If equals are added to equals, the wholes are equal."
  3. Question 3: Does Euclid's fifth postulate imply the existence of parallel lines? Explain.
    Solution: Yes. If the sum of interior angles on one side is exactly $180^\circ$, the lines will not meet on that side. If it is $180^\circ$ on the other side too, they never meet, implying they are parallel.
  4. Question 4: In the figure, if $AC = BD$, then prove that $AB = CD$.
    Solution: Given $AC = BD$.
    $AC = AB + BC$ and $BD = BC + CD$.
    So, $AB + BC = BC + CD$. Subtract $BC$ from both sides (Axiom 3).
    $\therefore AB = CD$.
  5. Question 5: Write Playfair's Axiom (equivalent to Euclid's 5th Postulate).
    Solution: "For every line $l$ and for every point $P$ not lying on $l$, there exists a unique line $m$ passing through $P$ and parallel to $l$."
  6. Question 6: Why is Axiom 5 ("The whole is greater than the part") considered a universal truth?
    Solution: Because it applies not just to geometry but to all magnitudes and quantities in the universe (e.g., mass, volume, numbers).
  7. Question 7: Prove that an equilateral triangle can be constructed on any given line segment.
    Solution: Let segment be $AB$. Draw circle with center $A$ radius $AB$, and circle with center $B$ radius $BA$. They intersect at $C$. $AC=AB$ and $BC=AB$ (radii). Thus $AC=BC=AB$.
  8. Question 8: If $AB = PQ$ and $PQ = XY$, then prove that $AB = XY$.
    Solution: According to Euclid's first axiom: "Things which are equal to the same thing are equal to one another." Since both equal $PQ$, $AB = XY$.
  9. Question 9: How many dimensions do a solid, a surface, and a line have respectively?
    Solution: Solid: 3 dimensions. Surface: 2 dimensions. Line: 1 dimension.
  10. Question 10: It is known that $x + y = 10$ and $x = z$. Show that $z + y = 10$.
    Solution: Since $x = z$, we can substitute $z$ for $x$ in the first equation. This uses the axiom "Things which are equal to the same thing are equal to one another" (substitution principle).
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