# Planimetrics - examples

Study plane measurements, including angles, distances, and areas. In other words - measurement and calculation of shapes in the plane. Perimeter and area of plane shapes.

1. Right triangle from axes
A line segment has its ends on the coordinate axes and forms with them a triangle of area equal to 36 sq. Units . The segment passes through the point ( 5,2). What is the slope of the line segment. ?
A meter pole perpendicular to the ground throws a shadow of 40 cm long, the house throws a shadow 6 meters long. What is the height of the house?
3. Is right-angled
Can a triangle with the sides of sqrt 3, sqrt 5 and sqrt 8 (√3, √5 a √8) be a right triangle?
4. Isosceles
Isosceles trapezium ABCD ABC = 12 angle ABC = 40 ° b=6. Calculate the circumference and area.
5. Two chords
There is a given circle k (center S, radius r). From point A which lies on circle k are starting two chords of length r. What angle does chords make? Draw and measure.
6. Three sides
Side b is 2 cm longer than side c, side a is 9 cm shorter than side b. The triangle circumference is 40 cm. Find the length of sides a, b, c . .. .
The double ladder is 8.5m long. It is built so that its lower ends are 3.5 meters apart. How high does the upper end of the ladder reach?
8. Square into three rectangles
Divide the square with a side length of 12 cm into three rectangles with have the same circumference so that these circumferences are as small as possible.
9. Right triangle
Calculate the length of the remaining two sides and the angles in the rectangular triangle ABC if a = 10 cm, angle alpha = 18°40'.
10. Two rectangles
I cut out two rectangles with 54 cm², 90 cm². Their sides are expressed in whole centimeters. If I put these rectangles together I get a rectangle with an area of 144 cm2. What dimensions can this large rectangle have? Write all options. Explain your calcu
11. Prove
Prove that k1 and k2 is the equations of two circles. Find the equation of the line that passes through the centers of these circles. k1: x2+y2+2x+4y+1=0 k2: x2+y2-8x+6y+9=0
12. Sum of inner angles
Prove that the sum of all inner angles of any convex n-angle equals (n-2) . 180 degrees.
13. The big clock
The big clock hands stopped at a random moment. What is the probability that: a) a small hand showed the time between 1:00 and 3:00? b) the big hand was in the same area as a small hand in the role of a)? c) did the hours just show the time between 21:00.
14. Sandbox
Sandbox has area of 32 sq ft and length of 4 1/2 ft. What is width of sandbox.
15. The perimeter
The perimeter of equilateral △PQR is 12. The perimeter of regular hexagon STUVWX is also 12. What is the ratio of the area of △PQR to the area of STUVWX?
16. Garden
Area of square garden is 6/4 of triangle garden with sides 56 m, 35 m and 35 m. How many meters of fencing need to fence a square garden?
17. Trapezoid MO
The rectangular trapezoid ABCD with right angle at point B, |AC| = 12, |CD| = 8, diagonals are perpendicular to each other. Calculate the perimeter and area of ​​the trapezoid.
18. Right Δ
Right triangle has length of one leg 51 cm and length of the hypotenuse 85 cm. Calculate the height of the triangle.
19. Hands
The clock shows 12 hours. After how many minutes will agle between hour and minute hand 90°? Consider the continuous movement of both hands hours.
20. River
Calculate how many promiles river Vltava average falls, if on section long 928 km flowing water from 1592 m AMSL to 108 m AMSL.

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