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Grade 10 Review Quiz — Easy Version 2
Grade 10 · MathSheets Name: ____________________ Date: __________
Answer every question. The lesson name before each question tells you which skill it checks — open that lesson if you need a reminder.
How to solve
This quiz mixes questions from different lessons, like a real test. Read each question carefully: the lesson name in front tells you what kind of problem it is. Check your answers, then practise the lessons where you made mistakes.
Example
Word Problems: A triangular sail has a base of 8 feet and a height of 5 feet. What is the area of the sail in square feet?
Answer: 20
Not quite. The answer is
💡 The volume of a pyramid is V = 13 × B × h. The base is a rectangle, so B = 4 × 3 = 12. V = 13 × 12 × 15 = 180 ÷ 3 = 60.
✓ Correct!
Not quite. The answer is
💡 The volume of a pyramid is V = 13 × B × h. The base is a rectangle, so B = 4 × 3 = 12. V = 13 × 12 × 15 = 180 ÷ 3 = 60.
💡 How to solve: Pyramids & Prisms: Volume and Surface Area
Volume of a prism = B · h and volume of a pyramid = 13 · B · h, where B is the area of the base and h is the height. For a square pyramid with base edge s and slant height l, lateral area = 12 · (4s) · l = 2sl and total surface area = s2 + 2sl. The height, half the base edge and the slant height form a right triangle.
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Find the volume of the square pyramid.
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💡 The exterior angles of any polygon add to 360°. A regular polygon has 6 equal exterior angles, so each one is 360° ÷ 6 = 60°.
✓ Correct!
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💡 The exterior angles of any polygon add to 360°. A regular polygon has 6 equal exterior angles, so each one is 360° ÷ 6 = 60°.
💡 How to solve: Polygon Angle Sums
The interior angles of an n-sided polygon add to (n - 2) × 180°. For a regular polygon, each interior angle = (n - 2) × 180° ÷ n and each exterior angle = 360° ÷ n. For a triangle subtract the known angles from 180°, and for a quadrilateral subtract them from 360°.
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Find the sum of the interior angles of a polygon with 4 sides.
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💡 Use the midpoint formula M = (x1 + x22, y1 + y22). x: -4 + 62 = 22 = 1. y: 1 + 12 = 22 = 1. The midpoint is (1, 1).
✓ Correct!
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💡 Use the midpoint formula M = (x1 + x22, y1 + y22). x: -4 + 62 = 22 = 1. y: 1 + 12 = 22 = 1. The midpoint is (1, 1).
💡 How to solve: Midpoint & Distance Formulas
For the midpoint, average the coordinates: midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). For the distance, use d = √((x₂ - x₁)² + (y₂ - y₁)²).
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Find the midpoint of the segment with endpoints A(-3, 0) and B(-3, 2).
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💡 Two pairs of angles are marked, ∠C ≅ ∠F and ∠A ≅ ∠D, and so is the side between them, AC ≅ DF. Two angles and the included side is ASA.
✓ Correct!
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💡 Two pairs of angles are marked, ∠C ≅ ∠F and ∠A ≅ ∠D, and so is the side between them, AC ≅ DF. Two angles and the included side is ASA.
💡 How to solve: Triangle Congruence: SSS, SAS, ASA, AAS & HL
List what is marked congruent, in order around the triangle. Three sides is SSS. Two sides and the angle between them is SAS. Two angles and the side between them is ASA; two angles and a side not between them is AAS. In right triangles, a hypotenuse and a leg is HL. Two sides and a non-included angle (SSA) or three angles (AAA) prove nothing, so the answer is none.
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Which postulate or theorem, if any, proves △ABC ≅ △DEF? (SSS, SAS, ASA, AAS, HL, or none)
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💡 The height x is opposite the 57° angle and the distance 29 ft is adjacent, so use tangent: tan 57° = x29. tan 57° ≈ 1.5399, so x = 29 × tan 57° ≈ 44.6561, which rounds to 44.7. The flagpole is about 44.7 ft tall.
✓ Correct!
Not quite. The answer is
💡 The height x is opposite the 57° angle and the distance 29 ft is adjacent, so use tangent: tan 57° = x29. tan 57° ≈ 1.5399, so x = 29 × tan 57° ≈ 44.6561, which rounds to 44.7. The flagpole is about 44.7 ft tall.
💡 How to solve: Angles of Elevation & Depression
An angle of elevation is measured up from the horizontal; an angle of depression is measured down from the horizontal, and it equals the angle of elevation from the other end (alternate interior angles). Sketch the right triangle, label the known side and the unknown, choose sin, cos or tan, and solve. Use inverse tangent (tan-1) to find an angle.
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You stand 17 ft from the base of a building. The angle of elevation to the top of the building is 60°. How tall is the building?
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💡 Corresponding sides of similar figures are proportional: 24 = x4. Cross-multiply: 4 × x = 2 × 4 = 8. Divide both sides by 4: x = 8 ÷ 4 = 2.
✓ Correct!
Not quite. The answer is
💡 Corresponding sides of similar figures are proportional: 24 = x4. Cross-multiply: 4 × x = 2 × 4 = 8. Divide both sides by 4: x = 8 ÷ 4 = 2.
💡 How to solve: Similar Figures
Corresponding sides of similar figures are in the same ratio. Set up a proportion, small side ÷ large side = x ÷ other large side, then cross-multiply and divide to solve for x.
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Two similar rectangles have a scale factor of 1:2. One side of the smaller rectangle is 3, and the corresponding side of the larger rectangle is 6. If another side of the larger rectangle is 8, what is the corresponding side of the smaller rectangle (x)?
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💡 The midpoint is the average of the endpoints: M = (a + b) ÷ 2 = (−3 + 9) ÷ 2 = 6 ÷ 2 = 3.
✓ Correct!
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💡 The midpoint is the average of the endpoints: M = (a + b) ÷ 2 = (−3 + 9) ÷ 2 = 6 ÷ 2 = 3.
💡 How to solve: Segment Addition & Midpoints
If B is between A and C, then AB + BC = AC. If M is the midpoint of AB, then AM = MB and AB = 2·AM. Write an equation from the given expressions, solve for x, and substitute back if a length is asked for. On a number line, the midpoint of a and b is a + b2.
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B is between A and C. AB = 9 and BC = 8. Find AC.
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💡 The two shorter lengths must add up to more than the longest. 4 + 12 = 16, and 16 > 13, so yes.
✓ Correct!
Not quite. The answer is
💡 The two shorter lengths must add up to more than the longest. 4 + 12 = 16, and 16 > 13, so yes.
💡 How to solve: Triangle Inequality & Side–Angle Order
Three lengths form a triangle only when the two shorter lengths add up to more than the longest. With two sides a and b, the third side x must satisfy (larger - smaller) < x < (larger + smaller). In any triangle, the largest angle is opposite the longest side and the smallest angle is opposite the shortest side.
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Can the lengths 4, 8, 7 be the side lengths of a triangle? Answer yes or no.
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💡 Slope of ℓ: From (−2, 5) to (−1, 7): rise = 7 − 5 = 2 and run = −1 − (−2) = 1. Slope = rise/run = 21 = 2. Perpendicular lines have opposite reciprocal slopes: the reciprocal of 2 is 12, and changing its sign gives −12.
✓ Correct!
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💡 Slope of ℓ: From (−2, 5) to (−1, 7): rise = 7 − 5 = 2 and run = −1 − (−2) = 1. Slope = rise/run = 21 = 2. Perpendicular lines have opposite reciprocal slopes: the reciprocal of 2 is 12, and changing its sign gives −12.
💡 How to solve: Slopes of Parallel & Perpendicular Lines
Parallel lines have equal slopes. Perpendicular lines have slopes that are opposite reciprocals (their product is -1), so a slope of 23 becomes -32. To write an equation, use the new slope m and the given point in y = mx + b, solve for b, then write y = mx + b.
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Line ℓ passes through A(-2, 0) and B(-1, 2). Find the slope of a line perpendicular to ℓ.
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💡 Opposite angles of a parallelogram are congruent. Angle C = angle A = 73°.
✓ Correct!
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💡 Opposite angles of a parallelogram are congruent. Angle C = angle A = 73°.
💡 How to solve: Quadrilateral Angle Relationships
In a parallelogram, consecutive angles add to 180°, so subtract the known angle from 180°, and opposite angles are equal. In any quadrilateral the four angles add to 360°, so subtract the three known angles from 360°.
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In parallelogram ABCD, angle A measures 12°. Angle A and angle B are consecutive angles. Find the measure of angle B.
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💡 The radius is half the diameter: 4 ÷ 2 = 2. Use A = πr2. A ≈ 3.14 × 22 = 3.14 × 4 = 12.56, which rounds to 12.6. The area is about 12.6.
✓ Correct!
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💡 The radius is half the diameter: 4 ÷ 2 = 2. Use A = πr2. A ≈ 3.14 × 22 = 3.14 × 4 = 12.56, which rounds to 12.6. The area is about 12.6.
💡 How to solve: Circles: Diameter & Advanced Problems
First find the radius: r = diameter ÷ 2. Circumference = π × d and area = π × r², using π ≈ 3.14. Round to one decimal place.
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A circle has a diameter of 4. Find its circumference. Use π ≈ 3.14.
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💡 By the Pythagorean Theorem, c2 = a2 + b2 = 52 + 122 = 25 + 144 = 169. So c = √169 = 13.
✓ Correct!
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💡 By the Pythagorean Theorem, c2 = a2 + b2 = 52 + 122 = 25 + 144 = 169. So c = √169 = 13.
💡 How to solve: Pythagorean Theorem
In a right triangle, a² + b² = c², where c is the hypotenuse. To find the hypotenuse, use c = √(a² + b²). To find a missing leg, use b = √(c² - a²).
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A right triangle has legs of length 7 and 24. Find the hypotenuse.
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💡 You know the side adjacent to θ and want the side opposite θ, so use tangent (TOA): tan θ = opposite/adjacent. tan 55° = x6. tan 55° ≈ 1.4281, so x = 6 × tan 55° ≈ 8.5689, which rounds to 8.6.
✓ Correct!
Not quite. The answer is
💡 You know the side adjacent to θ and want the side opposite θ, so use tangent (TOA): tan θ = opposite/adjacent. tan 55° = x6. tan 55° ≈ 1.4281, so x = 6 × tan 55° ≈ 8.5689, which rounds to 8.6.
💡 How to solve: Solving Right Triangles with Trigonometry
Label the sides relative to θ, then choose the ratio that links the side you know to the side you want: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent (SOH CAH TOA). Solve for the unknown by multiplying or dividing, and round to the nearest tenth.
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In a right triangle, angle θ = 55°, and the side adjacent to θ is 7. Find the length of the hypotenuse.
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💡 By the Inscribed Angle Theorem, an inscribed angle is half its intercepted arc: 38° ÷ 2 = 19°. The inscribed angle is 19°.
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💡 By the Inscribed Angle Theorem, an inscribed angle is half its intercepted arc: 38° ÷ 2 = 19°. The inscribed angle is 19°.
💡 How to solve: Circle Arcs & Inscribed Angles
A central angle has the same measure as its intercepted arc. An inscribed angle is half its intercepted arc, so the arc is twice the inscribed angle.
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A central angle intercepts an arc of 14°. Find the measure of the central angle.
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💡 By the Centroid Theorem, G is 23 of the way from A along median AD, so AG = 2 × GD. So GD = AG ÷ 2 = 14 ÷ 2 = 7.
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💡 By the Centroid Theorem, G is 23 of the way from A along median AD, so AG = 2 × GD. So GD = AG ÷ 2 = 14 ÷ 2 = 7.
💡 How to solve: Centroid, Circumcenter & Incenter
The centroid G (where the medians meet) is 23 of the way along each median, so AG = 2 · GD and AD = 3 · GD. The circumcenter (where the perpendicular bisectors meet) is the same distance from all three vertices. The incenter (where the angle bisectors meet) is the same distance from all three sides. Set the equal distances equal and solve.
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G is the centroid of △ABC and AG = 12. Find GD.
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💡 In (x − h)2 + (y − k)2 = r2, the number on the right is r2. So r2 = 9 and r = √9 = 3. The radius is 3.
✓ Correct!
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💡 In (x − h)2 + (y − k)2 = r2, the number on the right is r2. So r2 = 9 and r = √9 = 3. The radius is 3.
💡 How to solve: Equations of Circles
A circle with center (h, k) and radius r has equation (x - h)2 + (y - k)2 = r2. Watch the signs: (x + 3)2 means h = -3. The number on the right is r2, so take its square root for the radius. If the equation is expanded, group the x terms and the y terms and complete the square for each.
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Write the equation of the circle with center (5, 1) and radius 5.
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Grade 10 Review Quiz — Easy Version 2
Answer every question. The lesson name before each question tells you which skill it checks — open that lesson if you need a reminder.