All questions of Quadrilaterals for Class 9 Exam

Because sum of parallelogram is 180degree

The figure formed by joining the mid-points of consecutive sides of a quadrilateral is a Parallelogram.

Explanation:
A quadrilateral is a polygon with four sides. Let's consider a quadrilateral ABCD, where AB, BC, CD, and DA are the four sides.

Now, let's join the mid-points of the consecutive sides of the quadrilateral. Let the midpoints of AB, BC, CD, and DA be E, F, G, and H respectively.

To prove that the figure formed is a parallelogram, we need to show that opposite sides are parallel and equal in length.

1. Opposite sides are parallel:
- Join EF and GH. These diagonals divide the quadrilateral ABCD into four triangles: AEF, BFG, CGH, and DHG.
- By the Midpoint Theorem, EF is parallel to AB and GH is parallel to CD.
- Similarly, EG is parallel to AD and FH is parallel to BC.
- Therefore, opposite sides EF and GH are parallel, and opposite sides EG and FH are parallel.

2. Opposite sides are equal in length:
- By the Midpoint Theorem, EF = 1/2 AB and GH = 1/2 CD.
- Similarly, EG = 1/2 AD and FH = 1/2 BC.
- Therefore, opposite sides EF and GH are equal in length, and opposite sides EG and FH are equal in length.

Since the figure has opposite sides parallel and equal in length, it satisfies the definition of a parallelogram.

Hence, the figure formed by joining the mid-points of consecutive sides of a quadrilateral is a parallelogram.

Note: It is important to note that the converse of this statement is also true. That is, if a quadrilateral is a parallelogram, then the midpoints of its sides will form a parallelogram.

Diagonals in Different Figures:

In order to determine which of the given figures have equal diagonals, let's explore the properties of each figure.

a) Rectangle:
- A rectangle is a quadrilateral with four right angles.
- Opposite sides of a rectangle are parallel and congruent.
- Diagonals of a rectangle bisect each other, meaning they intersect at their midpoint.
- The diagonals of a rectangle are equal in length.

b) Parallelogram:
- A parallelogram is a quadrilateral with opposite sides that are parallel and congruent.
- The diagonals of a parallelogram bisect each other.
- However, the diagonals of a parallelogram are not necessarily equal in length.

c) Rhombus:
- A rhombus is a quadrilateral with all sides of equal length.
- Opposite sides of a rhombus are parallel.
- The diagonals of a rhombus bisect each other.
- Moreover, the diagonals of a rhombus are perpendicular to each other.
- The diagonals of a rhombus are not necessarily equal in length.

d) Trapezium:
- A trapezium is a quadrilateral with at least one pair of parallel sides.
- The diagonals of a trapezium do not necessarily bisect each other.
- Moreover, the diagonals of a trapezium are not necessarily equal in length.

Conclusion:

From the above analysis, we can conclude that the diagonals are equal in a rectangle but not in a parallelogram, rhombus, or trapezium. Therefore, the correct answer is option 'A' - Rectangle.

B)144

Let the four angles be 1x, 2x, 3x, and 4x respectively.

Since the sum of angles in a quadrilateral is 360 degrees, we have:

1x + 2x + 3x + 4x = 360
10x = 360
x = 36

Therefore, the largest angle is 4x = 4(36) = 144 degrees.

Given information:
- Length of each side of the rhombus = 15 cm
- Length of one diagonal = 24 cm

To find: Length of the other diagonal

Properties of a rhombus:
1. All sides of a rhombus are equal in length.
2. The diagonals of a rhombus bisect each other at right angles.

Using these properties, we can solve the problem.

Solution:
Let's denote the length of the other diagonal as 'd'.

Step 1: Find the length of the other side of the rhombus.
Since all sides of a rhombus are equal in length, the length of each side is 15 cm.

Step 2: Find the length of the diagonals.
The diagonals of a rhombus bisect each other at right angles. This means that they divide the rhombus into four congruent right-angled triangles.

Using the Pythagorean theorem, we can find the length of the diagonals.
In each right-angled triangle, the hypotenuse is the length of the diagonal, and the two legs are the lengths of the sides of the rhombus.

Applying the Pythagorean theorem:
(15/2)^2 + (d/2)^2 = (24/2)^2
225/4 + d^2/4 = 144
225 + d^2 = 576
d^2 = 576 - 225
d^2 = 351

Taking the square root of both sides:
d = √351
d ≈ 18.7 cm (rounded to the nearest tenth)

Step 3: Choose the correct option.
The question asks for the length of the other diagonal, so the answer is approximately 18.7 cm, which is closest to option (c) 18 cm.

Therefore, the correct answer is option (c) 18 cm.

C) Angle R.

There are several ways to approach this problem, but one of the most straightforward methods is to draw a square and label its sides and midpoints. Let's go through the solution step by step.

Step 1: Draw a square
Start by drawing a square with all sides of equal length. Label the four corners as A, B, C, and D.

Step 2: Label the midpoints
Next, label the midpoints of each side of the square. Let's call the midpoint on AB as E, BC as F, CD as G, and DA as H.

Step 3: Join the midpoints
Now, join the midpoints consecutively. That is, join E and F, F and G, G and H, and finally, H and E.

Step 4: Observe the resulting figure
Take a moment to observe the resulting figure formed by joining the consecutive midpoints. You will notice that it is a smaller square inside the original square.

Step 5: Identify the shape
Based on our observation, we can conclude that the resulting figure obtained from joining the consecutive midpoints of the sides of a square is another square. Therefore, the correct answer is option 'B' - Square.

Explanation:
When we join the consecutive midpoints of the sides of a square, we are essentially connecting the midpoints of each side. This creates a smaller square inside the original square. This smaller square shares the same center and orientation as the original square, but its sides are shorter in length. Therefore, the resulting figure is another square. This can be proven mathematically as well using properties of similar triangles and the fact that the diagonals of a square bisect each other at right angles.

In conclusion, the correct answer is option 'B' - Square.