Mathematics Questions

Browse 839 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

Q81
In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
1. (i)Find $E C$ , given $A D = 1.5$ cm, $D B = 3$ cm, $A E = 1$ cm (Fig. 6.17 (i)).
2. (ii)Find $A D$ , given $D B = 7.2$ cm, $A E = 1.8$ cm, $E C = 5.4$ cm (Fig. 6.17 (ii)).
Fig. 6.17 (ii)
Fig. 6.17 (i)
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Q82
E and F are points on the sides PQ and PR respectively of a triangle PQR. For each of the following cases, state whether EF || QR:
1. (i)PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
2. (ii)PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
3. (iii)PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
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Q83

In Fig. 6.18, if $L M ∥ C B$ and $L N ∥ C D$ , prove that $\frac{A M}{A B} = \frac{A N}{A D}$ .
Fig. 6.18

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Q84

In Fig. 6.19, $D E ∥ A C$ and $D F ∥ A E$ . Prove that $\frac{B F}{F E} = \frac{B E}{E C}$ .

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Q85

In Fig. 6.20, $D E ∥ O Q$ and $D F ∥ O R$ . Show that $E F ∥ QR$ .
Fig. 6.20

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Q86

In Fig. 6.21, $A, B$ and $C$ are points on $O P$ , $O Q$ and $O R$ respectively such that $A B ∥ P Q$ and $A C ∥ P R$ . Show that $B C ∥ QR$ .
Fig. 6.21

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Q87

Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

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Q88

Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

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Q89

ABCD is a trapezium in which $A B ∥ D C$ and its diagonals intersect each other at the point O. Show that $\frac{A O}{B O} = \frac{C O}{D O}$ .

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Q90

The diagonals of a quadrilateral $A B C D$ intersect each other at the point $O$ such that $\frac{A O}{B O} = \frac{C O}{D O}$ . Show that $A B C D$ is a trapezium.

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Q91
Find the distance between the following pairs of points:
1. (i)(2,3), (4,1)
2. (ii)(-5,7), (-1,3)
3. (iii)(a,b), (-a,-b).
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Q92

Find the distance between the points $(0, 0)$ and $(36, 15)$ . Can you now find the distance between the two towns A and B discussed in Section 7.2?

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Q93

Determine if the points (1, 5), (2, 3) and (-2, -11) are collinear.

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Q94

Check whether (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.

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Q95

In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, 'Don't you think ABCD is a square?' Chameli disagrees. Using distance formula, find which of them is correct.

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Q96
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
1. (i) $(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)$ ;
2. (ii) $(- 3, 5), (3, 1), (0, 3), (- 1, - 4)$ ;
3. (iii) $(4, 5), (7, 6), (4, 3), (1, 2)$ .
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Q97

Find the point on the $x$ -axis which is equidistant from $(2, - 5)$ and $(- 2, 9)$ .

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Q98

Find the values of $y$ for which the distance between the points $P (2, - 3)$ and $Q (10, y)$ is $10$ units.

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Q99

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

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Q100

Find a relation between $x$ and $y$ such that the point $(x, y)$ is equidistant from the point $(3, 6)$ and $(- 3, 4)$ .

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