Grade 10 Math

1. 1.Match each number to its prime factorisation.

   • 36
   • 60
   • 84
   • 90

   • 2² × 3 × 5
   • 2 × 3² × 5
   • 2² × 3²
   • 2² × 3 × 7

2. 2.Euclid's division lemma says that for whole numbers a and b there are whole numbers q and r with a = bq + r, where r is smaller than b. Divide 455 by 42. What is the remainder?

3. 3.√2 is neither a whole number nor a fraction, so it sits between two ticks on this line. Tap the tick just below it.

   Mark the line with an X.

4. 4.Find the HCF of 616 and 32 using Euclid's algorithm.

5. 5.Find the HCF of 6, 72 and 120.

6. 6.Find the LCM of 12 and 18.

7. 7.For any two numbers, HCF × LCM equals the product of the numbers. The HCF of two numbers is 9 and their LCM is 360. One of the numbers is 45. What is the other?

8. 8.13/125 has a decimal expansion that stops. After how many decimal places does it stop?

   • a) 2
   • b) 1
   • c) 4
   • d) 3

Real numbers · Grade 10 Math · www.arenapublications.com/learn

1. 1.Two right-angled triangles have equal hypotenuses and one pair of equal shorter sides. Which rule proves they are congruent?

   • a) RHS
   • b) ASA
   • c) SAS
   • d) SSS

2. 2.Triangle ABC is congruent to triangle PQR. Angle A is 55°. How big is angle P, in degrees?

3. 3.Two triangles have two pairs of sides equal, and the angle between those sides is equal too. Which rule proves they are congruent?

   • a) ASA
   • b) SAS
   • c) SSS
   • d) RHS

4. 4.Match each set of matching parts to the congruence rule it gives you.

   • Three pairs of sides
   • Two sides and the angle between them
   • Two angles and the side between them
   • Hypotenuse and one side, in right-angled triangles

   • ASA
   • RHS
   • SAS
   • SSS

5. 5.Triangle ABC is congruent to triangle PQR. AB is 7 inches long. How long is PQ, in inches?

6. 6.Two triangles have all three pairs of sides equal. Which rule proves they are congruent?

   • a) SAS
   • b) RHS
   • c) ASA
   • d) SSS

7. 7.Two triangles have two pairs of angles equal, and the side between those angles is equal too. Which rule proves they are congruent?

   • a) ASA
   • b) RHS
   • c) SSS
   • d) SAS

8. 8.Sort each set of matching parts by whether it is enough to prove congruence.

   Groups: Proves congruence · Does not prove congruence

   • AAA
   • SSA
   • SSS
   • ASA
   • SAS
   • RHS

Congruent triangles · Grade 10 Math · www.arenapublications.com/learn

1. 1.Two similar triangles have sides in the ratio on the left. Match it to the ratio of their areas.

   • 1:2
   • 1:3
   • 2:3
   • 3:4
   • 1:5

   • 1:25
   • 9:16
   • 1:9
   • 1:4
   • 4:9

2. 2.A 2 ft pole casts a shadow 3 ft long. At the same moment a tree casts a shadow 18 ft long. How tall is the tree, in feet?

3. 3.Two similar triangles have sides in the ratio 3:5. The perimeter of the smaller is 18 inches. What is the perimeter of the larger, in inches?

4. 4.The Basic Proportionality Theorem says that a line drawn parallel to one side of a triangle...

   • a) is equal in length to the side it is parallel to
   • b) makes the triangle isosceles
   • c) cuts the other two sides exactly in half
   • d) divides the other two sides in the same ratio

5. 5.Which of these is true of two similar triangles?

   • a) Matching sides are equal
   • b) They must be right-angled
   • c) They must be the same size
   • d) Matching sides are in the same ratio

6. 6.Two similar triangles have sides in the ratio 2:3. The area of the smaller is 16 square inches. What is the area of the larger, in square inches?

7. 7.Sort each pair of shapes by whether they must always be similar.

   Groups: Always similar · Not always similar

   • Two circles
   • Two isosceles triangles
   • Two squares
   • Two rectangles
   • Two rhombuses
   • Two equilateral triangles

8. 8.Are any two squares similar to each other?

   • a) No, never
   • b) Only if they are the same size
   • c) Only if they are drawn on grid paper
   • d) Yes, always

Similar triangles · Grade 10 Math · www.arenapublications.com/learn

1. 1.Sort each ratio by its value.

   Groups: Equals 0 · Equals 1

   • sin 0°
   • cos 90°
   • tan 0°
   • sin 90°
   • cos 0°
   • tan 45°

2. 2.What is sin 90°?

3. 3.Match each trigonometric ratio to its value.

   • sin 30°
   • cos 30°
   • tan 45°
   • sin 0°
   • tan 60°

   • √3/2
   • 1/2
   • 0
   • √3
   • 1

4. 4.What is tan 45°?

5. 5.In a right-angled triangle the side next to an angle is 4 inches and the hypotenuse is 5 inches. What is the cosine of that angle? Give your answer as a fraction.

6. 6.What is sin 60°?

   • a) 1/2
   • b) √3/2
   • c) √3
   • d) 1/√3

7. 7.What is sin 30°? Give your answer as a fraction.

8. 8.What is tan 30°?

   • a) 1/2
   • b) √3/2
   • c) 1/√3
   • d) √3

Introduction to trigonometry · Grade 10 Math · www.arenapublications.com/learn

1. 1.The Sun is 45° above the horizon. How long a shadow does an upright pole 12 ft tall cast on level ground, in feet?

2. 2.From the top of a cliff 40 ft high, the angle of depression of a boat at sea is 45°. How far is the boat from the base of the cliff, in feet?

3. 3.From the top of a cliff, the angle of depression of a boat is 30°. As the boat sails straight towards the base of the cliff, what happens to the angle of depression?

   • a) It increases
   • b) It becomes an angle of elevation
   • c) It decreases
   • d) It stays the same

4. 4.From a window you look down at a car, and from the car someone looks up at the window. How do the angle of depression and the angle of elevation compare?

   • a) They add up to 90°
   • b) The angle of elevation is larger
   • c) They are equal
   • d) The angle of depression is larger

5. 5.A kite is flying on a straight string 100 ft long that makes an angle of 30° with the level ground. How high is the kite above the ground, in feet?

6. 6.The angle of elevation of the top of a tower from a point on level ground is 60°, and the point is 45 ft from the base of the tower. Take √3 = 1.73. How tall is the tower, in feet?

7. 7.An upright pole casts a shadow. Put these angles of elevation of the Sun in order, shortest shadow first.

   • 60°
   • 75°
   • 30°
   • 45°

8. 8.A tree is 20 ft tall. From a point on level ground the angle of elevation of its top is 30°. Take √3 = 1.73. How far is that point from the base of the tree, in feet?

Heights and distances · Grade 10 Math · www.arenapublications.com/learn

1. 1.The point P divides the segment joining (1, 2) and (7, 11) in the ratio 2 : 1, measured from (1, 2). What is the y-coordinate of P?

2. 2.Which of these points is the same distance from the origin as (3, 4)?

   • a) (1, 6)
   • b) (2, 3)
   • c) (4, 4)
   • d) (5, 0)

3. 3.How far is the point (7, 24) from the origin? Give the answer in units.

4. 4.This line runs from 0 to 10. Tap the point that divides it in the ratio 2 : 3, measured from 0.

   Mark the line with an X.

5. 5.Find the area of the triangle whose vertices are (0, 0), (4, 0) and (0, 6). Give the answer in square units.

6. 6.Match each pair of points to the distance between them, in units.

   • (0, 0) and (5, 12)
   • (0, 0) and (9, 12)
   • (0, 0) and (8, 6)
   • (1, 1) and (4, 5)

   • 5
   • 10
   • 15
   • 13

7. 7.Find the area of the triangle whose vertices are (2, 1), (8, 1) and (2, 9). Give the answer in square units.

8. 8.What is the area of the triangle whose vertices are (1, 2), (2, 4) and (3, 6)? Give the answer in square units.

   • a) 2
   • b) 1
   • c) 3
   • d) 0

Distance and section formulas · Grade 10 Math · www.arenapublications.com/learn

1. 1.A tangent from a point P touches a circle with center O at A. The radius OA is 8 inches and the hypotenuse OP is 17 inches. How long is the tangent PA, in inches?

2. 2.Two tangents are drawn from a point P to a circle, touching it at A and B. How does PB compare with PA?

   • a) PB is twice PA
   • b) It depends on the radius
   • c) PB is the same length as PA
   • d) PB is half of PA

3. 3.A tangent from a point P touches a circle with center O at A. The radius OA is 5 inches and the hypotenuse OP is 13 inches. How long is the tangent PA, in inches?

4. 4.Match each circle and outside point to the length of the tangent from that point, in inches.

   • radius 3 inches, point 5 inches from the center
   • radius 8 inches, point 10 inches from the center
   • radius 12 inches, point 13 inches from the center
   • radius 6 inches, point 10 inches from the center

   • 6
   • 8
   • 4
   • 5

5. 5.OAPB is a quadrilateral with right angles at A and B, and its angle at P is 70°. What is its angle at O, in degrees?

6. 6.A tangent touches a circle with center O at the point P. What is the angle between the tangent and the radius OP?

   • a) 180°
   • b) 60°
   • c) 45°
   • d) 90°

7. 7.How many points does a tangent share with its circle?

   • a) Infinitely many
   • b) 0
   • c) 2
   • d) 1

8. 8.A tangent PA is 24 inches long and the hypotenuse OP of triangle OAP is 25 inches. How long is the radius OA, in inches?

Circles and tangents · Grade 10 Math · www.arenapublications.com/learn

1. 1.In a circle of radius 10 inches a sector has an angle of 90° at the center. Take π = 3.14. The triangle made by the two radii and the chord joining their ends has an area of 50 square inches. What is the area of the minor segment cut off by that chord, in square inches?

2. 2.A circle has been cut into three sectors. Tap the sector with the largest area.

   Write the number of the part.

3. 3.A sector of a circle of radius 6 inches has an angle of 60° at the center. Take π = 3.14. What is its area, in square inches?

4. 4.A sector of a circle of radius 21 inches has an angle of 60° at the center. Take π = 22/7. What is its area, in square inches?

5. 5.A sector of a circle of radius 21 inches has an angle of 60° at the center. Take π = 22/7. What is the perimeter of the sector, in inches?

6. 6.Doubling a sector's radius while keeping its angle at the center the same multiplies its area by what?

   • a) 4
   • b) 3
   • c) 2
   • d) 8

7. 7.A sector of a circle of radius 7 inches has an angle of 90° at the center. Take π = 22/7. How long is its arc, in inches?

8. 8.Through how many degrees does the minute hand of a clock turn in 20 minutes?

Areas related to circles · Grade 10 Math · www.arenapublications.com/learn

1. 1.A cylinder of radius 7 inches and height 10 inches has a cone of radius 7 inches and slant height 25 inches standing on top of it. Its outside is the cone's curved surface, the cylinder's curved surface and the circular base. Take π = 22/7. What is the total surface area, in square inches?

2. 2.A solid is a cylinder of radius 7 inches and height 10 inches with a cone of radius 7 inches and height 12 inches standing on top of it. Take π = 22/7. What is the total volume of the solid, in cubic inches?

3. 3.A solid is a cylinder of radius 7 inches and height 10 inches with a cone of radius 7 inches and height 12 inches standing on top of it. Take π = 22/7. What is the volume of the cone part, in cubic inches?

4. 4.Sort each formula by what it measures.

   Groups: A surface area · A volume

   • πrl
   • 1/3 πr²h
   • 2πrh
   • πr²h
   • 4/3 πr³
   • 3πr²

5. 5.A toy is a cone standing on a hemisphere, both of radius 7 inches. The cone is 24 inches tall. What is the cone's slant height, in inches?

6. 6.A solid is a cylinder of radius 7 inches and height 10 inches with a cone of radius 7 inches and height 12 inches standing on top of it. Take π = 22/7. What is the volume of the cylinder part, in cubic inches?

7. 7.A cone, a hemisphere and a cylinder all have a radius of 7 inches, and the cone and the cylinder are both 7 inches tall. Which holds the most?

   • a) The cone
   • b) The cylinder
   • c) The hemisphere
   • d) They all hold the same

8. 8.A sphere of radius 7 inches is cut exactly in half. Take π = 22/7. What is the total surface area of one hemisphere, counting its flat face, in square inches?

Combined solids · Grade 10 Math · www.arenapublications.com/learn