GSEB Class 8 Maths Solutions Chapter 3 ચતુષ્કોણની સમજ Exercise 3.4

 

Question 1. નીચેના વિધાનો સાચાં છે કે ખોટાં તે જણાવો.
(a) દરેક લંબચોરસ ચોરસ છે.
Answer: ખોટું. No, this statement is false. A rectangle has all its angles as right angles, but its sides are not always equal. For a shape to be a square, all its sides must be equal in length as well. Therefore, not every rectangle can be called a square.
In simple words: A rectangle has right angles but sides don't have to be equal. A square needs equal sides, so not all rectangles are squares.

Exam Tip: Remember that a square is a special type of rectangle, but a rectangle is not necessarily a square. Understanding the precise definitions of geometric shapes is key.

 

Question 1.
(b) દરેક સમબાજુ ચતુષ્કોણ સમાંતરબાજુ ચતુષ્કોણ છે.
Answer: ખરું. Yes, this statement is true. A rhombus has all four sides equal in length. This means its opposite sides are parallel and also equal. These properties fulfill the definition of a parallelogram.
In simple words: A rhombus has all sides equal, which means its opposite sides are parallel and equal too. So, it is always a parallelogram.

Exam Tip: Recall that a parallelogram requires opposite sides to be parallel and equal. Since a rhombus satisfies these, it's always a parallelogram.

 

Question 1.
(c) દરેક ચોરસ સમબાજુ ચતુષ્કોણ છે તેમજ લંબચોરસ પણ છે.
Answer: ખરું. Yes, this statement is true. A square has all four sides equal in length, which makes it a rhombus. Additionally, a square has all four right angles, which also makes it a rectangle. Therefore, a square holds the properties of both a rhombus and a rectangle.
In simple words: A square has equal sides (like a rhombus) and right angles (like a rectangle). So, a square is both a rhombus and a rectangle.

Exam Tip: Squares are unique as they possess properties of many other quadrilaterals, including rhombuses, rectangles, and parallelograms.

 

Question 1.
(d) દરેક ચોરસ સમાંતરબાજુ ચતુષ્કોણ નથી.
Answer: ખોટું. No, this statement is false. A square is actually a type of parallelogram because its opposite sides are parallel and also equal in length. Therefore, it fits all the requirements of a parallelogram.
In simple words: A square always has opposite sides parallel and equal, which means it definitely is a parallelogram.

Exam Tip: Understanding the hierarchy of quadrilaterals is important. A square inherits properties from parallelograms, rhombuses, and rectangles.

 

Question 1.
(e) દરેક પતંગાકાર ચતુષ્કોણ સમબાજુ ચતુષ્કોણ છે.
Answer: ખોટું. No, this statement is false. A kite has two distinct pairs of equal-length adjacent sides. However, for a figure to be a rhombus, all four of its sides must be equal in length. A kite typically does not have all four sides of the same measure, so it cannot be considered a rhombus.
In simple words: A kite only has two pairs of equal adjacent sides, not all four sides equal. So, a kite is not usually a rhombus.

Exam Tip: Differentiate between the properties: kites have equal adjacent sides, while rhombuses have all sides equal.

 

Question 1.
(f) દરેક સમબાજુ ચતુષ્કોણ પતંગાકાર ચતુષ્કોણ છે.
Answer: ખરું. Yes, this statement is true. A rhombus has all four sides equal in length. If all four sides are equal, then it automatically has two distinct pairs of equal-length adjacent sides (since any two adjacent sides will be equal). This property meets the definition of a kite.
In simple words: A rhombus has all sides equal. This means any two sides next to each other are equal, so it always counts as a kite.

Exam Tip: A rhombus is a special type of kite where all four sides are equal. This makes the statement true.

 

Question 1.
(g) દરેક સમાંતરબાજુ ચતુષ્કોણ સમલંબ ચતુષ્કોણ છે.
Answer: ખરું. Yes, this statement is true. A parallelogram has two pairs of parallel opposite sides. A trapezium (or trapezoid) is defined as a quadrilateral that has at least one pair of parallel sides. Since a parallelogram has two such pairs, it satisfies the condition of having at least one pair of parallel sides, making it a trapezium.
In simple words: A parallelogram has two sets of parallel sides. A trapezium only needs one set of parallel sides. So, every parallelogram is also a trapezium.

Exam Tip: The definition of a trapezium (at least one pair of parallel sides) is broad enough to include parallelograms.

 

Question 1.
(h) દરેક ચોરસ સમલંબ ચતુષ્કોણ છે.
Answer: ખરું. Yes, this statement is true. A square has two pairs of parallel opposite sides. Since a trapezium (or trapezoid) is defined as a quadrilateral with at least one pair of parallel sides, a square easily meets this requirement. Therefore, every square can also be classified as a trapezium.
In simple words: A square has parallel sides, and a trapezium needs only one set of parallel sides. So, a square is always a trapezium.

Exam Tip: Similar to parallelograms, squares also fulfill the broader definition of a trapezium due to their parallel sides.

 

Question 2. એવા ચતુષ્કોણનાં નામ આપો કે જેમાં :
(a) ચારેય બાજુની લંબાઈ સમાન હોય.
Answer: ચોરસ અને સમબાજુ ચતુષ્કોણની ચારેય બાજુઓની લંબાઈ સમાન હોય છે. Squares and rhombuses have all four sides of equal length.
In simple words: Squares and rhombuses are shapes where all four sides are exactly the same length.

Exam Tip: Clearly distinguish that 'all sides equal' is the key property for both squares and rhombuses.

 

Question 2.
(b) ચાર કાટખૂણા હોય.
Answer: ચોરસ અને લંબચોરસમાં ચારેય ખૂણા કાટખૂણા હોય છે. In squares and rectangles, all four angles are right angles.
In simple words: Squares and rectangles are the shapes that always have four perfect right angles.

Exam Tip: Right angles (90 degrees) define rectangles and squares; ensure your answer covers both.

 

Question 2.
(i) ચતુષ્કોણ
Answer: ચોરસને ચાર બાજુઓ છે તેથી તે ચતુષ્કોણ છે. A square has four sides, so it is a quadrilateral.
In simple words: A square is a quadrilateral because it has four sides.

Exam Tip: A quadrilateral is any polygon with four straight sides and four vertices. Squares fit this fundamental definition.

 

Question 2.
(ii) સમાંતરબાજુ ચતુષ્કોણ
Answer: ચોરસની સામસામેની બાજુઓ સમાન છે તથા સમાંતર છે તેથી તે સમાંતરબાજુ ચતુષ્કોણ છે. The opposite sides of a square are equal and parallel, so it is a parallelogram.
In simple words: A square is a parallelogram because its opposite sides are equal and run parallel to each other.

Exam Tip: To identify a parallelogram, look for two pairs of parallel sides. Squares satisfy this, so they are parallelograms.

 

Question 2.
(iii) સમબાજુ ચતુષ્કોણ
Answer: ચોરસની બધી બાજુઓ સમાન હોય છે તેથી તે સમબાજુ ચતુષ્કોણ છે. All sides of a square are equal, so it is a rhombus.
In simple words: Since all sides of a square are the same length, it is also a rhombus.

Exam Tip: A rhombus is defined by having all four sides of equal length. Squares inherently have this property.

 

Question 2.
(iv) લંબચોરસ છે તે વિગતવાર સમજાવો.
Answer: ચોરસના બધા ખૂણા કાટખૂણા છે તેથી તે લંબચોરસ છે. All the angles of a square are right angles, so it is a rectangle.
In simple words: A square has all right angles, which means it is also a rectangle.

Exam Tip: The defining feature of a rectangle is having four right angles. A square meets this criteria.

 

Question 4. નીચે દર્શાવ્યા મુજબ વિકર્ણ ધરાવતાં ચતુષ્કોણનાં નામ આપો.
(i) પરસ્પર દુભાગે
Answer: જેના વિકણ પરસ્પર દુભાગે છે તેવા ચતુષ્કોણ નીચે પ્રમાણે છે : The quadrilaterals whose diagonals bisect each other are as follows:

• સમાંતરબાજુ ચતુષ્કોણ (Parallelogram)
• લંબચોરસ (Rectangle)
• ચોરસ (Square)
• સમબાજુ ચતુષ્કોણ (Rhombus)

In simple words: The shapes where the diagonals cut each other exactly in half are parallelograms, rectangles, squares, and rhombuses.

Exam Tip: This property (diagonals bisecting each other) is common to all parallelograms and their special forms.

 

Question 4.
(ii) પરસ્પરના લંબદ્વિભાજક હોય
Answer: જેના વિકણા પરસ્પરના લંબદ્વિભાજક હોય તેવા ચતુષ્કોણ નીચે પ્રમાણે છે: The quadrilaterals whose diagonals are perpendicular bisectors of each other are as follows:

• ચોરસ (Square)
• સમબાજુ ચતુષ્કોણ (Rhombus)

In simple words: Only squares and rhombuses have diagonals that not only cut each other in half but also cross each other at a perfect right angle.

Exam Tip: Remember that 'perpendicular' (forming a 90-degree angle) is the additional condition here, narrowing down the list to squares and rhombuses.

 

Question 4.
(iii) સમાન હોય
Answer: જેના વિકણનાં માપ સમાન છે તેવા ચતુષ્કોણ નીચે પ્રમાણે છે : The quadrilaterals whose diagonals are equal in measure are as follows:

• ચોરસ (Square)
• લંબચોરસ (Rectangle)

In simple words: Squares and rectangles are the only shapes whose diagonals are always the same length.

Exam Tip: Equal diagonals are a defining property of rectangles and, by extension, squares.

 

Question 5. લંબચોરસ એક બહિર્મુખ ચતુષ્કોણ છે, સમજાવો.
Answer:

• લંબચોરસમાં દરેક ખૂણાનું માપ 180° કરતાં ઓછું છે. In a rectangle, the measure of each angle is less than 180°.
• બંને વિકણ લંબચોરસના અંદરના જ ભાગમાં હોય છે. તેથી લંબચોરસ એ બહિર્મુખ ચતુષ્કોણ છે. Both diagonals are contained entirely within the interior of the rectangle. Therefore, a rectangle is a convex quadrilateral.

In simple words: A rectangle is convex because all its angles are less than 180 degrees, and both of its diagonals stay inside the shape.

Exam Tip: A polygon is convex if all its interior angles are less than 180 degrees and all its diagonals lie entirely inside the polygon.

 

Question 6. કાટકોણ ત્રિકોણ ABCમાં કાટખૂણાની સામેની બાજુનું મધ્યબિંદુ O છે. શિરોબિંદુઓ A, B અને Cથી બિંદુ O કેવી રીતે સમાન અંતરે આવે છે તે સમજાવો. (અહીં તૂટક રેખાઓ તમારી સહાયતા માટે દોરેલ છે.)
Answer:
In the figure, extend \(\overrightarrow{BO}\) to D such that \(BO = OD\). Now, draw \(\overline{CD}\) and \(\overline{AD}\). Thus, ABCD is formed.

In quadrilateral ABCD, \(AO = OC\) (given, since O is the midpoint of AC) and \(BO = OD\) (by construction). Therefore, in ABCD, the diagonals bisect each other. So, ABCD is a parallelogram in which \(\angle B\) is a right angle (given). Hence, ABCD is a rectangle.

Now, the diagonals of a rectangle are equal in measure and bisect each other.
\( \implies AC = BD \) and \( AO = OC = BO = OD \). Thus, O is equidistant from A, B, and C.
In simple words: Point O is the middle of the longest side of a right triangle. If we make a rectangle from this triangle, O becomes the center where the two long lines cross. Because these lines in a rectangle are equal and cut each other in half, O is the same distance from all three corners of the original triangle.

Exam Tip: Remember the property that the midpoint of the hypotenuse of a right-angled triangle is equidistant from all three vertices. This point is also the circumcenter of the triangle.