Class 9 Maths: Introduction to Euclid’s Geometry – Extra Questions with Answers
Q. 1 How many dimensions does a line segment have?
Check Solution
Ans: B
Solution: A line segment is a one-dimensional object.
Answer: 1
Q. 2 The faces of a triangular pyramid are all ______ .
Check Solution
Ans: A
Solution: The faces of a triangular pyramid are all triangles.
Answer: triangles
Q. 3 If Thales is 5 years older than his sister, and his sister is currently 12 years old, how old is Thales?
Check Solution
Ans: C
Solution: Thales’ age = Sister’s age + 5 years
Answer: 17
Q. 4 A store sells apples for $0.75 each and oranges for $0.50 each. If Sarah buys 4 apples and 6 oranges, how much does she spend in total?
Check Solution
Ans: A
Solution:
Cost of apples: 4 * $0.75 = $3.00
Cost of oranges: 6 * $0.50 = $3.00
Total cost: $3.00 + $3.00 = $6.00
Answer: $6.00
Q. 5 A rectangular prism has a length of 8 cm, a width of 5 cm, and a height of 3 cm. What is the surface area of the prism?
Check Solution
Ans: C
Solution:
Surface Area = 2lw + 2lh + 2wh
Surface Area = 2(8)(5) + 2(8)(3) + 2(5)(3)
Surface Area = 80 + 48 + 30
Surface Area = 158 cm²
Answer: 158 cm²
Q. 6 The locus of a point equidistant from the points (1, 3) and (5, 3) intersects the line y = 1. What is the x-coordinate of the intersection point?
Check Solution
Ans: B
Solution: The locus of points equidistant from (1, 3) and (5, 3) is the perpendicular bisector of the segment connecting them. The midpoint is ((1+5)/2, (3+3)/2) = (3, 3). The segment is horizontal, so the perpendicular bisector is the vertical line x = 3. The intersection with y = 1 is (3, 1).
Answer: 3
Q. 7 What is the defining characteristic of a theorem that distinguishes it from a postulate?
Check Solution
Ans: A
Solution:
Answer: Theorems are proven, while postulates are assumed to be true.
Q. 8 Given a line segment $\overline{AB}$ of length 7 units and a point $C$ located 3 units from $A$ along $\overline{AB}$. If a line $l$ is drawn through $C$ perpendicular to $\overline{AB}$, and another line $m$ is parallel to $\overline{AB}$ such that the distance between $l$ and $m$ is 5 units, what is the area of the rectangle formed by $\overline{AB}$ and a segment of $m$ of the same length as $\overline{AB}$?
Check Solution
Ans: C
Solution:
The length of $\overline{AB}$ is 7 units. The distance between $l$ and $m$ is 5 units. The area of the rectangle is the product of the length and width, which is $7 \times 5 = 35$.
Answer: 35
Q. 9 If lines $l$ and $m$ intersect at a point $P$, and line $n$ passes through $P$ such that $l$ and $m$ are both perpendicular to $n$, what is the relationship between $l$ and $m$?
Check Solution
Ans: B
Solution:
Answer: $l$ and $m$ are the same line.
Q. 10 Find the equation of the locus of a point P equidistant from the lines $x + 2y – 3 = 0$ and $2x – y + 4 = 0$, given that P lies in the first quadrant.
Check Solution
Ans: C
Solution:
The distance from a point $(x, y)$ to the line $ax + by + c = 0$ is given by $\frac{|ax + by + c|}{\sqrt{a^2 + b^2}}$.
Let P be $(x, y)$. Then the distance from P to the line $x + 2y – 3 = 0$ is $\frac{|x + 2y – 3|}{\sqrt{1^2 + 2^2}} = \frac{|x + 2y – 3|}{\sqrt{5}}$.
The distance from P to the line $2x – y + 4 = 0$ is $\frac{|2x – y + 4|}{\sqrt{2^2 + (-1)^2}} = \frac{|2x – y + 4|}{\sqrt{5}}$.
Since P is equidistant from the two lines, we have $\frac{|x + 2y – 3|}{\sqrt{5}} = \frac{|2x – y + 4|}{\sqrt{5}}$. Thus $|x + 2y – 3| = |2x – y + 4|$.
We have two cases:
Case 1: $x + 2y – 3 = 2x – y + 4$, so $x – 3y + 7 = 0$.
Case 2: $x + 2y – 3 = -(2x – y + 4)$, so $3x + y + 1 = 0$.
Since P lies in the first quadrant, $x > 0$ and $y > 0$.
For Case 1, we have $x = 3y – 7$. Since $x > 0$ and $y > 0$, we have $3y – 7 > 0$, so $y > \frac{7}{3}$. Also, $x = 3y – 7 > 0$, and since we are in the first quadrant, $y>0$.
For Case 2, we have $y = -3x – 1$. However, since $x > 0$, we have $y < -1$, which is not in the first quadrant.
Thus, the locus of P is $x – 3y + 7 = 0$, with $y > \frac{7}{3}$.
Answer: $x – 3y + 7 = 0$, $y > \frac{7}{3}$
Next Topic: Lines and Angles
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