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Geometry

1. In a quadrilateral OABC, D is the midpoint of BC and E is the point on AD such

that AE : ED = 2 : 1. Given that OA = a, OB = b and OC = c, express OD and

OE in terms of a, b and c.

Geometry

In a quadrilateral OABC, D is the midpoint of BC and E is the point on AD such that AE : ED = 2 : 1. Given that OA = A, OB = B, and OC = c express OD and OE in terms of a,b and c.

Geometry

m ∠ R X Q = ( x + 15 ) , m ∠ R X S = ( 5 x − 7 ) , m ∠ Q X S = ( 3 x + 5 ) .

ABC is a triangle and P, Q are the midpoints of AB, AC respectively. If AB = 2x

and AC = 2y, express the vectors (i) BC, (ii) PQ, (iii) PC, (iv) BQ in terms of x

and y. What can you deduce about the directed line-segments BC and PQ?

Geometry

Let BCB′C′ be a rectangle, let M be the midpoint of B′C′, and let A be a point on the circumcircle of the rectangle. Let triangle ABC have orthocenter H, and let T be the foot of the perpendicular from H to line AM. Suppose that AM= 2, [ABC] = 2020, and BC= 10. Then AT=m/n, where m and n are positive integers with gcd (m,n) = 1. Compute 100m+n.

Can two rectangles have:

a.

equal areas and unequal perimeters?

b.

equal perimeters and unequ

al areas?

c.

equal perimeters and unequal diagonals?

d.

equal diagonals, and unequal perimeters?

e.

equal diagonals, equal bases, and unequal perimeters?

f.

equal diagonals, equal altitudes, and unequal areas?

Express:

a.

the altitude h of an equilateral triangle as a function of its side s

b.

the hypotenuse c of a r

ight triangle as a function of side s.

c.

a side s of a square as a function of its diagonal d.

d.

the median m of a trapezoid as a function of its area A and its altitude h.

Geometry

Find the length of the arc on the unit circle with the given central angles

a. 315º

b. 240º

Formulate at least two problems invovlimg arcs and central angles,thensolve

Geometry

ABCD is a square and P, Q are the midpoints of BC, CD respectively. If AP = a

and AQ = b, find in terms of a and b, the directed line segments (i) AB