Q1. Which of the following sets of lengths form a right triangle?
Q2. In $\Delta ABC$, $AB = 6\sqrt{3}$ cm, $AC = 12$ cm and $BC = 6$ cm. The angle B is:
Q3. The length of the diagonal of a square is $10\sqrt{2}$ cm. Its side is:
Q4. A ladder 10 m long reaches a window 8 m above the ground. The distance of the foot of the ladder from the base of the wall is:
Q5. Assertion (A): In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Reason (R): This statement is known as Pythagoras Theorem.
Q6. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
Solve on white paper.
Q7. $ABC$ is an isosceles triangle right angled at $C$. Prove that $AB^2 = 2AC^2$.
Q8. Find the length of the altitude of an equilateral triangle of side $2a$.
Q9. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
Q10. In an equilateral triangle $ABC$, $D$ is a point on side $BC$ such that $BD = \frac{1}{3}BC$. Prove that $9AD^2 = 7AB^2$.
Q11. Case Study: Airplanes
An airplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another airplane leaves the same airport and flies due west at a speed of 1200 km per hour.
(i) How far does the first plane travel in $1 \frac{1}{2}$ hours?
(ii) How far does the second plane travel in $1 \frac{1}{2}$ hours?
(iii) How far apart will be the two planes after $1 \frac{1}{2}$ hours?
Q12. Case Study: Guy Wire
A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end.
(i) How far from the base of the pole should the stake be driven so that the wire will be taut?
(ii) If the wire length is increased to 30 m, how far from the base should the stake be driven?
(iii) Which mathematical concept is used to solve this problem?