Converse of Pythagoras Theorem

Q.

Find the perimeter of rectangle whose length is $40$ cm and diagonal is $41$ cm.

Q. The sides of a triangle are of length 1.5 cm, 2.5 cm and 2 cm. Then this triangle is a right triangle.

1. False
2. True

Q. The sides of a triangle are of length 4 cm, 7 cm and 6 cm. Then this triangle is a right triangle.

1. True
2. False

Q. Use the sign of $>,<$ or $=$ in the box to make the statement true.
$-231+79+51\square -399+159+81$

Q. Find the perimeter of a rectangle if its diagonal is 5 cm and the smaller side is 3 cm as shown in the figure below.

Q. The sides of a triangle are of lengths 8 cm, 15 cm and 17 cm. The triangle is a triangle.

1. right angled
2. acute angled
3. obtuse angled

Q.

The blades of a helicopters propeller are perpendicular and congruent.

The distance between consecutive blade tips is $24feet$.

Find the length $\left(b\right)$ of each blade.

Q.

Two poles of height 10 cm and 20 cm stand on a plane ground. If the distance between their feet is 24 cm. Find the distance between their tops (in cm).

1. 26

Q.

Lengths of sides of a triangle are $3cm,4cm$ and $5cm$. The triangle is

1. Obtuse angled triangle

2. Acute-angled triangle

3. Right-angled triangle

4. An Isosceles right triangle

Q.

In a triangle with side lengths a, b, and c, where c is the length of the longest side, then choose the condition for acute triangle.

1. c${}^2$>a${}^2$+b${}^2$

2. c${}^2$=a${}^2$+b${}^2$

3. c${}^2$<a${}^2$+b${}^2$

Q. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is right-angled triangle.

Q. If $\overrightarrow{v_1}and\overrightarrow{v_2}$ are two vectors in x – y plane. Then any vector in that plane can be obtained by the linear combination of these two vectors.The statement is

1. True
2. False

Q. The sides of certain triangles are given below. Determine which of them are right triangles
(i) a = 7 cm, b = 24 cm and c= 25 cm
(ii) a = 9 cm, b = 16 cm and c = 18 cm

Q. Hypotenuse of a right triangle is $25cm$ and out of the remaining two sides, one is longer than the other by $5cm$. Find the lengths of the other two sides.

1. $10$cm and $20$ cm
2. $15$cm and $20$ cm
3. $25$cm and $20$ cm
4. $5$cm and $20$ cm

Q.

Question 7
Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41cm.

Q. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is right-angled triangle.

Q.

If the sides of a triangle are 26 cm, 20 cm and 15 cm, then the triangle is right angled.

Q.

If the sides of a triangle are 50 cm, 40 cm and 30 cm, then the triangle is right angled.

Q. Let $\overrightarrow{a}=2\overrightarrow{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}$ and a unit vector $\overrightarrow{c}$ be coplanar. If $\overrightarrow{c}$ is perpendicular to $\overrightarrow{a},$ then $\overrightarrow{c}$ is equal to

1. $\frac{1}{\sqrt{2}}(-\hat{j}+\hat{k})$
2. $\frac{1}{\sqrt{3}}(-\hat{i}-\hat{j}-\hat{k})$
3. $\frac{1}{\sqrt{5}}(-\hat{i}-2\hat{j})$
4. $\frac{1}{\sqrt{5}}(\hat{i}-\hat{j}-\hat{k})$

Q. Let, $\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}.$
A vector coplanar to $\overrightarrow{a}$ and $\overrightarrow{b}$ has a projection along $\overrightarrow{c}$ of magnitude $\frac{1}{\sqrt{3}}$, then the vector is

1. $4\hat{i}-\hat{j}+4\hat{k}$
2. $4\hat{i}+\hat{j}-4k$
3. $2\hat{i}+\hat{j}+\hat{k}$
4. None of these

Q. The sides of a triangle are of length 8 cm, 15 cm and 17 cm. Then this triangle is a right triangle.

1. True
2. False

Q. If $\overrightarrow{v_1}and\overrightarrow{v_2}$ are 2 vectors, then $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}.\overrightarrow{v_2}$ is the linear combination of these 2 vectors where $c_1,c_2\stackrel{`}{\epsilon }R$.

1. True
2. False

Q. The linear combination of $\overline{a}=\left[\begin{array}{c}1\\ 2\end{array}\right]and\overline{b}=\left[\begin{array}{c}0\\ 3\end{array}\right]$ which gives the vector $\left[\begin{array}{c}2\\ 2\end{array}\right]$ will be ___$\overline{a}$ +__ $\overline{b}$. (If any of the blanks is a fraction, enter the value to the nearest hundredth place).

Q.

Given three side of a triangle can be the sides of a right angled triangle or not?
2.5 cm, 6.5 cm, 6cm
In case of right angled triangle, identify the right angle.
[2 marks]

Q. Prove: $(+3)+(-7)=(-4)$

Q. In the triangle ABC, given below , AB$=$8 cm, BC$=$ 6 cm and AC $=$ 3 cm. Calculate length of OC.

1. $5.6cm$
2. $2.6cm$
3. $3.6cm$
4. $1.6cm$