Pythagoras Theorem

Q.

In the given figure, $\angle ABC={90}^o$ and $BD\perp AC$. If BD = 8 cm, AD = 4 cm, find CD.

Q. A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

Q.

All theorems in the REAL NUMBERS

Q.

Let $P$ be an $3\times 3$ matrix $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$ where $\alpha$ belongs to $\mathrm{ℝ}$. Suppose $Q=\left[q_{ij}\right]$ is a matrix $PQ=k{I}_3$for some non-zero $k\in \mathrm{ℝ}$. If $q_{23}=-k/8$ and $\left|Q\right|=k^2/2$, then ${\alpha }^2+k^2$ is equal to _________.

Q.

$\Delta ABC$ is an equilateral triangle of side 2a units. Find each of its altitudes.

Q.

State the converse of Pythagoras' theorem.

Q.

State whether the given statement is true or false:

8, 15, 17 is a Pythagorean triplet

1. True

2. False

Q.

What is the inradius of an equilateral triangle, the length of whose sides is $2\sqrt{3}$ cm?

1. 1 cm

2. 2 cm

3. 3 cm

4. 4 cm

Q. If p₁ and p₂ are two odd prime numbers such that p₁ > p₂, then $p_1^2-p_2^2$ is

(a) an even number
(b) an odd number
(c) an odd prime number
(d) a prime number

Q. The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus.

Q. The perpendicular from A on side BC of a $\Delta ABC$ intersects BC at D such that DB = 3CD .Prove that $2A{B}^2=2A{C}^2+B{C}^2$.

Q.

State whether the given statement is true or false:

9, 40, 41 is a Pythagorean triplet.

1. False

2. True

Q.

ABC is a right triangle right angled at B such that BC = 6 cm and AB = 8 cm. The radius of its in circle is

1. 8 cm

2. 2 cm

3. 5 cm

4. 3 cm

Q.

The sum of two vectors $A$ and $B$ is at right angle to their difference. Then,

1. $A=B$

2. $A=2B$

3. $B=2A$

4. $A$ and $B$ have the same direction

Q.

In a right $\Delta ABC$ right-angled at C, if D is the mid-point of BC, prove that $B{C}^2=4(A{D}^2-A{C}^2)$.

Q.

Which of the following cannot be the sides a right triangle?

1. 9 cm, 5 cm, 7cm

2. 400mm, 300 mm, 500mm

3. 9 cm, 15 cm, 12 cm

4. 2 cm, 1 cm, cm

Q. Prove that in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angles opposite to the first side is a right angle.

Q. Which of the following are not the sides of a right angled triangle?

1. 5, 12, 13
2. 6, 8, 10
3. 10, 24, 25
4. 7, 24, 25

Q. ∆ABC is an isosceles triangle in which ∠C = 90. If AC = 6 cm, then AB =

(a) $6\sqrt{2}\mathrm{cm}$
(b) 6 cm
(c) $2\sqrt{6}\mathrm{cm}$
(d) $4\sqrt{2}\mathrm{cm}$

Q.

$ABC$ is an isosceles triangle right angled at $C$. Prove that $A{B}^2=2A{C}^2$

Q. In a right triangle ABC, right angled at B, D is a point on hypotenuse such that BD⊥AC. If DP⊥AB and DQ⊥BC then
prove that
(a) DQ² = DP.QC
(b) DP² = DQ.AP

Q. Question 6
Prove that a triangle must have atleast two acute angles.

Q.

Find the area of the quadrilateral ABCD in which AD = 24 cm, $\angle$ BAD = ${90}^o$ and $\Delta$ BCD is an equilateral triangle having each side equal to 26 cm. Also, find the perimeter of the quadrilateral. [Given, $\sqrt{3}$ = 1.73.]

Q. In ∆ABC, ∠A = 60°. Prove that BC² = AB² + AC² − AB . AC.

Q.

In a right angled triangle, the longest side is the __.

Q. If ABC is an isosceles triangle and D is a point of BC such that AD ⊥ BC, then

(a) AB² − AD² = BD.DC
(b) AB² − AD² = BD² − DC²
(c) AB² + AD² = BD.DC
(d) AB² + AD² = BD² − DC²

Q.

State Pythagoras' theorem.

Q.

The lengths of the diagonals of a rhombus are $40cm$ and $42cm.$ Find the length of each side of the rhombus.

Q.

Question 5
A 5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4m high. If the foot of the ladder is moved 1.6m towards the wall then find the distance by which the top of the ladder would slide upwards on the wall.

Q.

In an isosceles right triangle, the length of the hypotenuse is 10 cm. Then the perimeter of the triangle is:

1. $5\sqrt{2}cm$

2. $2\sqrt{2}cm$

3. $10(\sqrt{2}+1)cm$

4. $10(\sqrt{2}-1)cm$