Dragan

Angle BAC ≈ 39 degrees

Two right-angled triangles 30-40-50 put together to form an isosceles triangle ABC.  

(x+4)² + (x+2)² = 10²

^{x = 4}

5x + 3 +5x - 5 = 78

x = 8

angle DBC = 5x + 3 = 43º

angle ABC = 5x - 5 = 35º

Angles:  PQR, PRQ, RTS, and RST are congruent. QR = 8  and  SQ = 2. Find PQ.

Hint:     QR = QT = 8     Place a point U on PQ so that TU || QR       TU = SQ = 2

PQ = 9.237604306    

20x - 10 + 14x - 19 = 90

x = 3.5

∠DAB = 180 - ( 14x - 19 ) = 150

1/  Chose the length of QR

2/  Find angles UPQ  and  UQP

3/  Calculate side QU of a triangle PQU

4/  Find angle VQU

5/  UV = sin(VQU) * QU = 12

Question:   How many different rectangles can be drawn on an 8*8 chessboard?

Answer:     8+7+6+5+4+3+2+1 = 36

36 different rectangles ( squares included ) can be drawn on a chessboard.  

A right triangle is inscribed in a circle with a diameter 100 units long. What is the maximum area of the triangle, in square units?

The largest right-angled triangle inscribed in a circle is an isosceles right triangle.

area = 100² / 4

(1) Let WZYZ be a trapezoid with bases XY and Z. In this trapezoid,angle ZXW = 105,angle XWZ = 43 , and angle XYZ = 141. Find angle YXZ in degrees.

1/     Angle YXZ = 32 degrees.

(2) In trapezoid EFGH, line EF is parallel to line GH  and  P is the midpoint of side EH. If the area of triangle PEF is 8, and the area of triangle PGH  is 21, then find the area of trapezoid EFGH.

2/    The area of a trapezoid EFGH is 58 units squared.

( I used the right trapezoid with side lengths: EF = 8,  GH = 21, and EH = 4 )

(3) Quadrilateral ABCD  is a parallelogram. Let E  be a point on AB, and let F be the intersection of lines DE and BC. The area of triangle EBF is 1 and the area of triangle EAD  is 9 Find the area of parallelogram ABCD.

3/   Area of parallelogram ABCD = 24 u²

 (4) In trapezoid  ABCD,  AB is parallel to CD, and  AB < CD. The base CD  has a length of 8 The legs  AD and BC have lengths of 7 and the diagonal BD  has a length of 9 Find the area of the trapezoid.

4/    Area = 6√45