Areas Related to Circles Assertion–Reason PYQs Class 10 CBSE (With Answers)

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Here r = 14 cm and θ = 90°.
Area of sector = $\frac{\text{θ}}{\text{360}}$ × πr².
= $\frac{\text{90}}{\text{360}}$ × $\frac{\text{22}}{\text{7}}$ × 14 × 14
= 154 cm².

Reason: True.
A sector of angle θ occupies $\frac{\text{θ}}{\text{360}}$ of the circle.
Its area is $\frac{\text{θ}}{\text{360}}$ × πr².

Link: R gives the formula used to calculate the area in A, so R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Let the radius be r cm.
Circumference = 2πr.
176 = 2 × $\frac{\text{22}}{\text{7}}$ × r.
r = $\frac{\text{176 × 7}}{\text{44}}$ = 28 cm.

Reason: True.
The circumference of a circle of radius r is 2πr.

Link: R gives the relationship used to find the radius in A, so R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Here the arc length is 22 cm and r = 21 cm.
Let the central angle be θ°.
Arc length = $\frac{\text{θ}}{\text{360}}$ × 2πr.
22 = $\frac{\text{θ}}{\text{360}}$ × 2 × $\frac{\text{22}}{\text{7}}$ × 21.
θ = $\frac{\text{22 × 360}}{\text{132}}$ = 60°.

Reason: True.
An arc subtending θ degrees occupies $\frac{\text{θ}}{\text{360}}$ of the full circumference.
Therefore its length is $\frac{\text{2πrθ}}{\text{360}}$.

Link: R gives the arc-length formula used to obtain 60°, so R explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
The area of a sector is $\frac{\text{θ}}{\text{360}}$ × πr².
The printed expression $\frac{\text{θ}}{\text{360}}$ × 2πr gives arc length, not area.

Reason: True for the corresponding minor segment.
Join the centre to the ends of its chord.
The minor sector consists of that segment and the triangle.
Hence area of segment = area of sector − area of triangle.

Link: A uses an arc-length formula for an area. R states the sector-minus-triangle result. A is false and R is true. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
Let the original radius be r and the original central angle be θ degrees.
Original arc length = $\frac{\text{πrθ}}{\text{180}}$.
New radius = $\frac{\text{r}}{\text{2}}$ and new angle = 2θ.
New arc length = $\frac{\text{π × (r/2) × 2θ}}{\text{180}}$ = $\frac{\text{πrθ}}{\text{180}}$.
Original perimeter = 2r + $\frac{\text{πrθ}}{\text{180}}$.
New perimeter = r + $\frac{\text{πrθ}}{\text{180}}$.
The perimeter decreases by r; it does not remain the same.

Reason: True.
Arc length = $\frac{\text{θ}}{\text{360}}$ × 2πr = $\frac{\text{πrθ}}{\text{180}}$.

Link: R correctly gives arc length. Perimeter also includes the two radii, whose total length changes from 2r to r. A is false and R is true. Answer: D.