Mathematics
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Fluidité · Pack B

9.1 Surds and Pythagoras

Répondez à chaque question. Montrez les calculs si nécessaire.

Bronze

  1. A right-angled triangle has the two shorter sides of length 6 cm and 8 cm. Find the hypotenuse.

  2. Simplify $\sqrt{72}$ fully.

  3. Simplify $\sqrt{45}$ fully.

  4. A triangle has sides 7, 24, 25 cm. Use Pythagoras' converse to decide whether it is right-angled.

  5. Simplify $\sqrt{5} \times \sqrt{20}$.

  6. Identify which numbers are rational vs irrational: $5, \sqrt{2}, \pi, 0.\overline{3}, \sqrt{4}$.

  7. Estimate $\sqrt{50}$ to 1 d.p.

  8. Find the hypotenuse of a right-angled triangle with legs 9 and 12 cm.

  9. Add: $\sqrt{50} + \sqrt{18}$ (give simplified form).

  10. A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other side.

Silver

  1. Pythagoras with surds: legs $2\sqrt{3}$ and 3. Find the hypotenuse.

  2. Find the distance between $(1, 2)$ and $(4, 6)$. Give answer as a surd in simplest form.

  3. Simplify $\sqrt{8} + \sqrt{32} - \sqrt{18}$.

  4. Find the hypotenuse of a right-angled triangle with legs 5 and 10 cm. Give answer as a simplified surd.

  5. Simplify $\sqrt{75} \div \sqrt{3}$.

  6. A right-angled triangle has hypotenuse $\sqrt{50}$ and one leg $\sqrt{18}$. Find the other leg.

  7. Find the distance between $(-2, 1)$ and $(3, 13)$.

  8. Find the perimeter of a right-angled triangle with legs 6 and 8 cm.

  9. Square the surd: $(2 + \sqrt{3})^2$. Expand.

  10. Rationalise the denominator: $\dfrac{6}{\sqrt{3}}$.

Gold

  1. A right-angled triangle has shorter sides 5 and 10. Hypotenuse as a simplified surd.

  2. Find the length of the space diagonal of a cuboid with sides 8, 6, 4. Use 3D Pythagoras.

  3. Pythagoras in 3D: cuboid 12 × 9 × 6. Find the space diagonal.

  4. A square has diagonal 10 cm. Find its side length and area.

  5. A ladder of length 10 m leans against a wall. Foot 3 m from the base. Find the height reached.

  6. Simplify $(3 + \sqrt{2})(2 - \sqrt{2})$.

  7. The area of a square is 12 cm². Find the side as a simplified surd.

  8. Find the length of the diagonal of a rectangle 6 cm by 8 cm.

  9. Show that the triangle with sides 5, 5, $5\sqrt{2}$ is right-angled.

  10. A ladder of length 8 m makes a 60° angle with the ground. Find the height up the wall (using trig).

Platinum

  1. A cuboid has length 8 cm, width 6 cm, height 4 cm. Find the angle between the space diagonal and the base, to 1 d.p.

  2. Make $a$ the subject of $c = \sqrt{a^2 + b^2}$.

  3. A cliff of height $h$ m stands on level ground. From point A 28 m from the base, the line of sight to the top is 52 m. From B further away, sight to top is 60 m. Find $h$ and B's distance.

  4. Apply Pythagoras to find a diagonal of a square pyramid: base side 8, apex height 12 above the centre. Find the slant distance from a base vertex to the apex.

  5. Rationalise: $\dfrac{1}{1 + \sqrt{2}}$.

  6. An isosceles triangle has equal sides 13 cm and base 10 cm. Find (a) the perpendicular height, (b) the area.

  7. Find the area of an equilateral triangle of side $s = 10$ cm.

  8. A right-angled triangle has legs $a$ and $b$ with $a + b = 14$ and $a^2 + b^2 = 100$. Find $a$ and $b$.

  9. Two ladders lean against opposite walls of an alley. One reaches 6 m up the left wall; the other 8 m up the right wall. The alley is 5 m wide. They cross at some height $h$. Find $h$ (use similar triangles).

  10. A triangle has sides $\sqrt{8}$, $\sqrt{18}$, $\sqrt{32}$. Show it is right-angled.