## CSETMathGuru: THE Site for Single Subject Math

**Are you prepared for Subtest II Geometry?**

Then, have a stab at the following!

1. A rectangle is to be inscribed in a semicircle of radius r cm. If the height of the rectangle is h, write an expression in terms of r and h for the Area and Perimeter of the rectangle. What dimensions of the rectangle yield the maximum Area?

2. ABC is an isosceles triangle inscribed in a circle with centre, O. (Assume A is the top vertex, B the vertex on the bottom right, C, on the bottom left. Also, AC = AB.) Further, AD, the diameter of the circle through O, meets CB at E. If AC = sqrt(15/2) cm and OE = 1 cm, estimate the radius of the circle. (Note: sqrt(15/2) ~ Square Root (15/2))

3. Find the area of the largest equilateral triangle that would fit

*inside*a circle of circumference 36

*cm.*

**pi**4. Find the area of the largest circle that would fit

*inside*an equilateral triangle with area 36 cm.

5. Find the perimeter of a square with an area same as that of a circle with circumference 30cm.

6. Find the area of a circle with perimeter same as that of a square with diagonal 10 cm.

7. ABCD is a rectangle with AB & CD as its long sides. It is enclosed in a circle with centre, O. The measure of angle AOB is 120 degrees and the length of AB is 12 cm. Find the Area of the region

*inside the circle*but

*outside the rectangle*.

8. A rectangle is inscribed inside a semi-circle of diameter 8 cm so that two of its vertices lie symmetrically on the semi-circle with the other two vertices on its diameter. If the area of the rectangle is sqrt(7), find the dimensions of the rectangle. Note: sqrt(7) ~ Square root (7)

9. Three

*congruent*circles are arranged to touch each other tangentially (ie. each touches the other two at one point only). A rectangle is drawn to circumscribe the 3 circles (ie. its sides act as tangents to the circles). If the radius of each circle is 1 cm, find the area and perimeter of the rectangle.

10. Imagine an Isosceles Triangle inscribed in a Semi-circle, with the longest side obviously the diameter. Now, using each congruent side of the triangle as base (ie. diameter) 2 smaller semicircles are drawn. Find the

*ratio*of the Area of the Triangle to the Sum of the Areas of the 2 semi-circles that lie

*outside*the large semi-circle.

11. A square is inscribed inside an equilateral triangle such that 2 of its vertices lie on 1 side of the triangle, and the other 2 vertices lie ON the other 2 sides of the triangle. Find the

*ratio*of the

*sum of the areas of the 2 triangles*(on the bottom, to the right & left of the square) to

*the area of the triangle on top*(the equilateral triangle on top of the square). Using the figure above, if the area of the square is 3, what is the perimeter of the triangle?

12. What does the following construction PO produce? Justify.

Step 1. Draw a horizontal line,

*l*, and a point, P, above it.

Step 2. With P as centre, draw an arc (Arc 1) cutting line,

*l*, at points A and B.

Step 3. With A as centre and radius > AB, draw a wide arc (Arc 2) on the other side of line,

*l*, across P.

Step 4. With B as centre and keeping the radius the same, draw an arc (Arc 3) cutting Arc 2 at O.

Step 5. Join PO.

13. Water flows into a tank 150m long and 100m wide through a pipe whose cross section is 0.2m by 0.15m at a speed of 15km per hour. In what time will the water be 3 meters deep?

14. An isosceles triangle sits in such a way that neither of its congruent sides is its base. Prove that the bisector of the exterior angle of its top vertex is parallel to its base.

15. In triangle ABC, BO and CO are the internal bisectors of Angle B and Angle C. Prove that Angle BOC = 90+ ½ Angle A.

16. A 12-sided regular convex polygon of side 10 cm is circumscribed by a circle.

a) Find the Area of the region inside the circle but outside the polygon.

b) If another circle were inscribed into the above polygon, find the Area of the region outside the circle but lying inside the polygon.