CBSE Class 10 · Mathematics

Geometry and Trigonometry

53 practice questions, each with the answer and an explanation.

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In $△ A B C$ , $D E ∥ B C$ such that $D$ is on $A B$ and $E$ is on $A C$ . If $A D = 3 cm$ , $D B = 5 cm$ , and $A E = 4.5 cm$ , what is the length of $E C$ ?
A line segment $A B$ has endpoints $A (- 2, 8)$ and $B (- 6, - 4)$ . What are the coordinates of the midpoint of $A B$ ?
From an external point $P$ , two tangents $P A$ and $P B$ are drawn to a circle with centre $O$ . If $∠ A P B = 7 0^{\circ}$ , what is the measure of $∠ A O B$ ?
If the perimeter and the area of a circle are numerically equal, what is the radius of the circle?
What is the distance between the points $P (2, - 3)$ and $Q (10, 3)$ ?
If $tan θ + cot θ = 2$ , where $0^{\circ} < θ < 9 0^{\circ}$ , what is the value of $tan^{7} θ + cot^{7} θ$ ?
A card is drawn from a well-shuffled standard pack of 52 playing cards. What is the probability of drawing either a king or a red queen?
The ratio of the areas of two similar triangles $△ A B C$ and $△ P QR$ is $64 : 121$ . If the median of $△ A B C$ is $12 cm$ , what is the corresponding median of…
In a right triangle $A B C$ right-angled at $B$ , if $sin A = 3/5$ , what is the value of $cos A$ ?
In $△ A B C$ , $D E ∥ B C$ with $D$ on $A B$ and $E$ on $A C$ . If $A D = 3 cm$ , $D B = 4 cm$ , and $A E = 6 cm$ , find the length of $E C$ .
If $\triangle ABC \sim \triangle PQR$ such that $Area (△ A B C) / Area (△ P QR) = 9/16$ and $B C = 4.5 cm$ , find the length of $QR$ .
A vertical pole of length 6 m casts a shadow 4 m long on the ground, and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
In an equilateral triangle $A B C$ , $D$ is a point on side $B C$ such that $B D = \frac{1}{3} B C$ . Prove that $9 A D^{2}$ equals what in terms of side $A B$ ?
In $△ A B C$ , $∠ B = 9 0^{\circ}$ and $D$ is the midpoint of $B C$ . Then $A C^{2} - A D^{2}$ is equal to:
If $S$ is a point on side $P Q$ of $△ P QR$ such that $P S = 3 cm$ and $S Q = 6 cm$ , and $T$ is on $P R$ such that $P T = 2 cm$ and $T R = 4 cm$ , which statement is tr…
The areas of two similar triangles $A B C$ and $D E F$ are $64 cm^{2}$ and $121 cm^{2}$ respectively. If $E F = 15.4 cm$ , find $B C$ .
In a right triangle $A B C$ right-angled at $C$ , if $p$ is the length of perpendicular from $C$ to $A B$ and $a, b, c$ are sides, which relation is correct?
Diagonals of a trapezium $A B C D$ with $A B ∥ D C$ intersect each other at $O$ . If $A B = 2 C D$ , what is the ratio of areas of $△ A O B$ and $△ C O D$ ?
Find the distance between the points $P (- 6, 7)$ and $Q (- 1, - 5)$ .
Find the coordinates of the point which divides the join of $(- 1, 7)$ and $(4, - 3)$ in the ratio $2 : 3$ internally.
If the points $A (6, 1)$ , $B (8, 2)$ , $C (9, 4)$ , and $D (p, 3)$ are the vertices of a parallelogram taken in order, find the value of $p$ .
Find the ratio in which the $y$ -axis divides the line segment joining the points $(5, - 6)$ and $(- 1, - 4)$ .
For what value of $k$ are the points $A (2, 3)$ , $B (4, k)$ , and $C (6, - 3)$ collinear?
Find the center of a circle passing through the points $(6, - 6)$ , $(3, - 7)$ , and $(3, 3)$ .
The line segment joining the points $A (3, 2)$ and $B (5, 1)$ is divided at the point $P$ in the ratio $1 : 2$ and it lies on the line $3 x - 18 y + k = 0$ . Find the value of $k$ .
Find the area of a rhombus if its vertices are $(3, 0)$ , $(4, 5)$ , $(- 1, 4)$ , and $(- 2, - 1)$ taken in order.
What point on the $x$ -axis is equidistant from $(2, - 5)$ and $(- 2, 9)$ ?
If $tan θ = \frac{4}{3}$ , what is the value of $\frac{sin θ + cos θ}{sin θ - cos θ}$ ?
Evaluate: $\frac{5 cos ^{2} 6 0 ^{\circ} + 4 sec ^{2} 3 0 ^{\circ} - tan ^{2} 4 5 ^{\circ}}{sin ^{2} 3 0 ^{\circ} + cos ^{2} 3 0 ^{\circ}}$ .
If $sin (A - B) = 1/2$ and $cos (A + B) = 1/2$ where $0^{\circ} < A + B \leq 9 0^{\circ}$ and $A > B$ , find $A$ and $B$ .
Prove the identity: $(sin θ + csc θ)^{2} + (cos θ + sec θ)^{2}$ simplifies to:
If $tan θ + sin θ = m$ and $tan θ - sin θ = n$ , which relation holds between $m$ and $n$ ?
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from $3 0^{\circ}$ to $6 0^{\circ}$ as he walks towards the…
As observed from the top of a 75 m high lighthouse from sea level, the angles of depression of two ships are $3 0^{\circ}$ and $4 5^{\circ}$ . If one ship is exactly behind the other on the same side of the…
If $sec θ + tan θ = x$ , what is the value of $tan θ$ in terms of $x$ ?
Evaluate: $sin 2 5^{\circ} cos 6 5^{\circ} + cos 2 5^{\circ} sin 6 5^{\circ}$ .
The angle of elevation of a cloud from a point $h$ meters above a lake is $α$ and the angle of depression of its reflection in the lake is $β$ . The height of the cloud above the lake surface…
From an external point $P$ , two tangents $P A$ and $P B$ are drawn to a circle with center $O$ . If $∠ A P B = 8 0^{\circ}$ , find $∠ A O B$ .
A quadrilateral $A B C D$ is drawn to circumscribe a circle. Which of the following equations must hold?
A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the minor sector. (Use $π = 3.14$ )
Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Find the area of the sector of a circle with radius 6 cm if angle of the sector is $6 0^{\circ}$ . (Use $π = 22/7$ )
If the perimeter and the area of a circle are numerically equal, then the radius of the circle is:
In a circle of radius 21 cm, an arc subtends an angle of $6 0^{\circ}$ at the center. Find the length of the arc. (Use $π = 22/7$ )
A tangent $P Q$ at a point $P$ of a circle of radius 5 cm meets a line through the center $O$ at a point $Q$ so that $O Q = 12 cm$ . Length of $P Q$ is:
A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.
Two cubes each of volume $64 cm^{3}$ are joined end to end. Find the surface area of the resulting cuboid.
A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere.
If the mean and median of a frequency distribution are 28 and 26 respectively, find the mode using the empirical relationship.
What is the probability that a leap year selected at random will contain 53 Sundays?
A card is drawn from a well-shuffled deck of 52 cards. What is the probability of getting a king of red color?
A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag.
Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is 8?

Geometry and Trigonometry

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