Length Based Approach

Q.

The length of a minute hand of a wall clock is 10.5 cm. The area swept by it in 10 minutes would be -

1. 5.77 cm²

2. 51.75 cm²

3. 57.75 cm²

4. 51.25 cm²

Q. ABC is an isosceles triangle and a circle such that it passes through vertex C and AB acts as a tangent at D for the same circle. AC and BC intersects the circle at E and F respectively AC = BC = 4 cm and AB = 6 cm. Also, D is the mid-point of AB. What is the ratio of EC : (AE + AD)?

1. 1 : 2
2. 1 : 3
3. 2 : 5
4. None of these

Q. Euclid has a triangle in mind. Its longest side has length 20 and another of its sides has length 10. Its area is 80. What is the exact length of its third side?

1. $\sqrt{240}$
2. $\sqrt{260}$
3. $\sqrt{270}$
4. $\sqrt{250}$

Q. Two persons start walking on roads that makes an angle of ${120}^{\circ }$ with each other. If they walk at the rate of 3 km/h and 2 km/h respectively. Find the distance between them after 4 hours.

1. 5 km
2. $4\sqrt{19}km$
3. $8\sqrt{19}km$
4. 7 m

Q. D, E, F are midpoints of BC, CA and AB respectively. G, H, I are midpoints of FE, FD, DE respectively. Areas of $\Delta DHI$ and $\Delta AFE$ are in the ratio

1. 1 : 3
2. 1 : 4
3. 1 : 9
4. 1 : 16

Q. In the given figure, ACB is a right angled triangle. CD is the altitude. Circles are inscribed within the triangles ACD, BCD. P and Q are the centres of the circles. The sum of the in radii of ACD and BCD is ___

Q. In the figure given below, the boundary of the shaded region comprises of four semicircles two-quarter circles. If OA = OB = OC = OD = 7 cm and the straight lines AC and BD are perpendicular to each other, find the length of the boundary.

1. 68 cm
2. 49 cm
3. 66 cm
4. 44 cm

Q.

What is the perimeter of a quadrant of a circle with radius $3cm$?

Q. In ABC, produce a line from B to AC, meeting at D, and from C to AB, meeting at E. Let BD and CE meet at X.Let $\Delta$ BXE have area a, $\Delta$ BXC have area b, and $\Delta$ CXD have area c. Find the area of quadrilateral AEXD in terms of a, b, and c.

1. $\frac{bc(2a+b+c)}{b^2-ac}$
2. $\frac{ac(a+2b+c)}{b^2-ac}$
3. $\frac{abc}{b^2-ac}$
4. $\frac{ab(a+b+2c)}{b^2-ac}$

Q.

Consider a square track ABCD. A circular track EPHQ is inscribed in the above square track, while a hexagonal track EFGHIJ is inscribed in the above circle and track EGI is a triangular path inscribed inside the hexagonal track. All figures are regular.
Four runners W, X, Y and Z start running simultaneously from point E on square, circular, hexagonal and triangular tracks respectively, but W and Y in clockwise direction and X and Z in anti-clockwise direction and they all meet for the first time when W, X, Y and Z have completed 1, 2, 3 and 4 rounds respectively. What is the ratio of their speeds?

1. $2:\pi :4:3\sqrt{3}$
2. $1:2\pi :9:4\sqrt{3}$
3. $4:2\pi :9:6\sqrt{3}$
4. $4:4\pi :9:12\sqrt{3}$

Q.

Consider a square track ABCD. A circular track EPHQ is inscribed in the above square track, while a hexagonal track EFGHIJ is inscribed in the above circle and track EGI is a triangular path inscribed inside the hexagonal track. All figures are regular.
If a, b , c and d are lengths of the square, circular, hexagonal and triangular tracks respectively, then which of the following is true?

1. $3\sqrt{3b}=\sqrt{3}\pi c=2d$
2. $3\pi a=12b=4\pi c$
3. $3\pi a=\sqrt{3}\pi c=2\pi d$
4. $3\pi a=12b=2\pi d$

Q. Triangle PQR is an equilateral with sides of length 5 units. O is any point in the interior of triangle PQR. Segment OL, OM, and ON are perpendicular to PQ, QR and PR respectively. Find the sum of segments OL, OM and ON.

1. Data insufficient
2. $\frac{10\sqrt{3}}{2}$
3. $\frac{3\sqrt{3}}{2}$
4. $\frac{5\sqrt{3}}{2}$

Q. Consider the five points comprising the vertices of a square and the intersection point of its diagonals. How many triangles can be formed using these points? (CAT 1993)

1. 6
2. 8
3. 4
4. 10

Q.

In the figure given below O is the center of the circle. ABCD can never be a -

1. Trapezium

2. Quadrilateral

3. Square

4. Rhombus

Q. In the given figure, ACB is a right angled triangle. CD is the altitude. Circles are inscribed within the triangles ACD, BCD. P and Q are the centres of the circles. The sum of the in radii of ACD and BCD is ___