Distance between Two Points Using Pythagoras Theorem

Q.

Write the formula for the perpendicular distance of a point from a line.

Q. If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.

Q.

Prove that the points (a, b + c), (b, c + a) and (c, a + b) are collinear.

Q.

If the distance between the points A (4, p) and B(1, 0) is 5 then

(a) p = 4 only (b) p = -4 only (c) p = $\pm$4 (d) p = 0

Q.

The centre of the circle passing through the point $(0,1)$ and touching the curve $y=x^2$ at $\left(2,4\right)$ is

1. $\left(\frac{3}{10},\frac{16}{5}\right)$

2. $\left(\frac{6}{5},\frac{53}{10}\right)$

3. $\left(\frac{-16}{5},\frac{53}{10}\right)$

4. $\left(\frac{-53}{10},\frac{16}{5}\right)$

Q.

The unit vector perpendicular to $\hat{i}-\hat{j}$and coplanar with$\hat{i}+2\hat{j}$and$\hat{i}+3\hat{j}$ is?

1. $\frac{2\hat{i}-5\hat{j}}{\sqrt{59}}$

2. $2\hat{i}+5\hat{j}$

3. $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

4. $\hat{i}+\hat{j}$

Q.

The base PQ of two equilateral triangles PQR and PQR' with side 2a lies along y-axis such that the mid-point of PQ is at the origin. Find the coordinates of the vertices R and R' of the triangles.

Q.

For a triangle ABC, which of the following is true ?

1. - =

2. AB – AC = BC

3. (AB – AC) > BC

4. (AB – AC) < BC

Q.

The least and the greatest distance of the point $(10,7)$ from the circle $x^2+y^2+4x-2y-20=0$ are

1. $10,5$

2. $15,20$

3. $12,16$

4. $5,15$

Q.

Show that the points $A\left(-2,3\right),B\left(8,3\right)\mathrm{and}C\left(6,7\right)$ are the vertices of a right triangle.

Q. Find the coordinates of the point(s) on x-axis which are at distance of 13 units from the point A(4, 5).

1. (1, 0) and (-8, 0)
2. (16, 0) and (-8, 0)
3. (8, 0) and (-1, 0)
4. (-6, 0) and (8, 0)

Q.

An ellipse passing through $(4\surd 2,2\surd 6)$ has foci at $(-4,0)$ and $(4,0)$. Then, its eccentricity is

1. $\sqrt{2}$

2. $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

3. $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.$

4. $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.$

Q. A line segment is of length 10 units. If the coordinates of its one end are (2, −3) and the abscissa of the other end is 10, then its ordinate is

(a) 9, 6

(b) 3, −9

(c) −3, 9

(d) 9, −6

Q.

The length of the hypotenuse in the figure is:

1. A and B

2. A and C

3. B and C

4. B and A

Q.

If a square with side ‘a’ is inserted within a circle such that the corners coincide with the circumference of the circle with diameter ‘d’. Find the relation between ‘a’ and ‘d’.

1. a = d

2. a = d/ √2

3. a = d/2

4. a = 2d

Q. In right angled triangle ABC, angle B = 90 degree , 'D' is the mid point of BC .prove that AC2=4AD2-3AB2

Q. Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (−2, −1), (1, 0), (4, 3) and(1, 2) meet.

Q.

Prove that the points (2, – 2), (–3, 8) and (–1, 4) are collinear.

Q.

For a triangle ABC, which of the following statements is true?

1. = +

2. AC < AB + BC

3. AC > AB + BC

4. AC = AB + BC

Q.

What is the length of the diagonal AG in the cuboid shown?

1. $\sqrt{1}04$

2. $\sqrt{1}4$

3. $\sqrt{2}8$

4. $2\sqrt{1}25$

Q.

If $Q(0,1)$ is equidistant from $P(5,-3)$ and $R(x,6)$, find the values of $x$.

Also find the distance $QR$ and $PR$.

Q.

For any triangle ABC, which of the following statements will always hold true?

1. AC > AB + BC

2. AC = AB + BC

3. = +

4. AC < AB + BC

Q. Show that the points A(a, a), B(-a, -a) and C(-a√3, a√3) form an equilateral triangle

Q. Find the condition that the points (a, b), (b, a) and (a², – b²) are in a straight line.

1. a + b = 0
2. a = b
3. a + b = 1
4. a + b = -1

Q. List-II gives distance between pair of points given in List-I, match them correctly.
$\begin{array}{cccc}& List-I& & ListII\\ \left(P\right)& (-5,7),(-1,3)& \left(1\right)& 5\\ \left(Q\right)& (5,6),(1,3)& \left(2\right)& \sqrt{8}\\ \left(R\right)& (\sqrt{3}+1,1),(0,\sqrt{3})& \left(3\right)& \sqrt{6}\\ \left(S\right)& (0,0),(-\sqrt{3},\sqrt{3})& \left(4\right)& 4\sqrt{2}\end{array}$

1. P-4, Q-3, R-2, S-1
2. P-4, Q-1, R-2, S-3
3. P-3, Q-2, R-4, S-1
4. P-1, Q-2, R-3, S-4

Q. Find the value of 'c' if the distance between the point (c, 4) and the origin is 5 units and c > 0.

1. -3
2. 3
3. 4
4. 5

Q. In Q. No. 14, write the sign of c.

Q.

In the given figure, ABC is a right-angled triangle, right angle is at B. Find lengths AB and BC and hence AC.

1. 4 unit

2. 8 unit

3. $4\sqrt{2}$ unit

4. $3\sqrt{2}$ unit

Q. 29. three vertices of a parallelogram are A (-2, 2), B(6, 2)and C(4, 3).plot these points on a graph paper and hence use it to find the co ordinates of the fourth vertex D . Also , find the coordinate of the mid point of the CD .

Q.

The length of the hypotenuse in the figure is.

1. 13
2. 5
3. 12