Geometry Problems PDF

## Geometry Problems **1) The hypotenuse of a right triangle is 41 cm, and its area is 180 cm². Find the legs of the triangle.** - 9 cm, 40 cm - 11 cm, 39 cm - 7 cm, 41 cm - 10 cm, 43 cm - 13 cm, 44 cm **2) One leg of a right triangle is 10 dm, the radius of the circle drawn outside the triangle...

## Geometry Problems **1) The hypotenuse of a right triangle is 41 cm, and its area is 180 cm². Find the legs of the triangle.** - 9 cm, 40 cm - 11 cm, 39 cm - 7 cm, 41 cm - 10 cm, 43 cm - 13 cm, 44 cm **2) One leg of a right triangle is 10 dm, the radius of the circle drawn outside the triangle is 13 dm. Find the area of the triangle.** - 120 dm² - 122 dm² - 123 dm² - 121 dm² - 124 dm² **3) Which method is the most effective for checking whether a point belongs to a parabola?** - Substituting the coordinates of the point into the equation of the parabola - Constructing a tangent at the given point - Finding the distance from the point to the focus - Constructing the medians of the parabola - Calculating the area of the figure **4) What is needed to find the equation of a hyperbola through its vertices?** - Determine the semi-axes of the hyperbola - Construct the circumscribed circle - Calculate the angles of inclination of the asymptotes - Draw the heights of the figure - Plot the graph of the figure **5) In which coordinates is it easier to solve the equations of an ellipse?** - In canonical coordinate system - In polar coordinates - In spherical coordinates - In Cartesian coordinates without transformation - In cylindrical system **6) The area of a right triangle is 150, one of its legs is 15. Find the length of the altitude drawn from the vertex of the right angle.** - 12 - 100√3 - 20 - 200√3 - 24 **7) Find the ratio of the legs of a right triangle whose area and one leg are 28 cm, 490 cm², and 490 cm², respectively.** - 4:5 or 5:4 - 3:4 or 4:3 - 1:2 or 2:1 - 1:3 or 3:1 - 2:3 or 3:2 **8) If the legs of a right triangle are cm and cm, find the area of this triangle.** - 65 cm² - 130 cm² - 120 cm² - 75 cm² - 55 cm² **9) The hypotenuse of a right triangle is 10 cm, and the radius of the inscribed circle is 2 cm. Find the area of the triangle.** - 24 cm² - 21 cm² - 27 cm² - 22 cm² **10) If the side of an isosceles triangle is 9 cm, find its area.** - 81√3/4 cm² - 3√3/2 cm² - 64√3/13 cm² - 16√3 cm² - 19√3 cm² **11) If the side of an isosceles triangle is 1 cm, find its area.** - √3/4 cm² - 3√3/2 cm² - 5√3/2 cm² - 9√3/4 cm² - 4√3 cm² **12) If the side of an isosceles triangle is 0.5 cm, find its area.** - √3/16 cm² - √3/4 cm² - 16√3 cm² - 9√3/2 cm² - 16√3 cm² **13) The side of a right triangle is cm. Find its area.** - 25√3/2 cm² - 9√3/4 cm² - 4√3 cm² - 6√3 cm² - 6√3 cm² **14) The area of a circle inscribed in a right triangle is 16π cm². Find the area of this triangle.** - 48√3 cm² - 44√3 cm² - 42√3 cm² - 45√3 cm² - 40√3 cm² **15) If the radius of a circle drawn outside a right triangle is cm, find the area of the triangle.** - 36√3 cm² - 36√2 cm² - 18√3 cm² - 48√3 cm² - 18√2 cm² **16) What is the side of an isosceles triangle inscribed in a circle of radius 10?** - 10√3 - 10 - 9 - 11√3 - 12√3 **17) The side of an isosceles triangle is 4 cm. Find its area.** - 4√3 cm² - 9√3/4 cm² - 4√3 cm² - 4√3/11 cm² - 6√3 cm² - 4√3/9 cm² **18) If the perimeter of an isosceles triangle is 10 cm, find the radius of the circle drawn outside it.** - 10√3/9 cm - 15 cm - 9√3 cm - 10√3 cm - 9√3/10 cm **19) The vertex A of an isosceles triangle ABC is connected to point D, dividing side BC into segments BD=1cm and DC=2 cm. Determine the length of segment AD.** - 7√7 cm - 11√11 cm - 6√6 cm - 13√13 cm - 5√5 cm **20) The median of an isosceles triangle is . Calculate the perimeter of the triangle.** - 90 - 60 - 45 - 120 - 75 **21) A right triangle is inscribed inside a circle with a radius of 20 cm. Find the length of the bisector of this triangle.** - 30 cm - 25 cm - 32 cm - 36 cm - 34 cm **22) If the side of a right triangle is 7 cm, find its perimeter.** - 21 cm - 25 cm - 19 cm - 18 cm - 24 cm **23) If the area of a right triangle is 16, find its perimeter.** - 24 cm - 27 cm - 32 cm - 36 cm - 28 cm **24) If the median of a right triangle is 1.5 cm, find its perimeter.** - 3√3 cm - 6√3 cm - 12√3 cm - 9√3 cm - 18 cm **25) If the perimeter of a right triangle is 30 cm, find its area.** - 25√3 cm² - 24√3 cm² - 32√3 cm² - 21√3 cm² - 20√3 cm² **26) If the perimeter of an equilateral triangle is 12 cm, find its altitude.** - 4√3 cm - 6√3 cm - 2√3 cm - 8√3 cm - 5√3 cm **27) If the height of an isosceles triangle is 45 cm, find the radius of the inscribed circle.** - 15 cm - 13 cm - 12 cm - 9 cm - 16 cm **28) If the height of an isosceles triangle is 20 cm, find the radius of the inscribed circle.** - 6√3/2 cm - 5√3/3 cm - 6√3/3 cm - 5√3/2 cm - 4√3/3 cm **29) If the height of an isosceles triangle is 16 cm, find the radius of the inscribed circle.** - 5√3/3 cm - 6√3/2 cm - 6√3/3 cm - 4√3/3 cm - 5√3/2 cm **30) If the bisector of an isosceles triangle is cm, find its area.** - √3 cm - 9√3/4 cm - 4√3/11 cm - 6√3 cm - 3√3/2 cm **31) If the median of an isosceles triangle is 6 cm, find its side.** - 4√3 cm - 5√2 cm - 4√2 cm - 5√3 cm - 3√3/4 cm **32) If the median of an isosceles triangle is cm, find its side.** - 9√3 cm - 5√3 cm - 4√3/4 cm - 3√3/11 cm - 6√3 cm **33) If the side of an isosceles triangle is 12 cm, find the radius of the circle drawn outside it.** - 4√3 cm - 7√2 cm - 7√3/6 cm - 6√3 cm - 3√3/4 cm **34) The side of an isosceles triangle is 6cm. Find its altitude.** - 3√3 cm - 2√3 cm - 7√6 cm - 4√6 cm - 7√3 cm **35) The side of an isosceles triangle is 5 cm. Find its bisector.** - 5√3/2 cm - 2√5 cm - 5√3/4 cm - 5√2 cm - 5√2/6 cm **36) The side of an isosceles triangle is 10 cm. Find its median.** - 5√3 cm - 7√2 cm - 12√2 cm - √3/2 cm - 2√3 cm **37) If the height of an isosceles triangle is 8 cm, find its side.** - 16√3/3 cm - 5√7 cm - 4√3 cm - 5√6/4 cm - 5√3 cm **38) If the area of an isosceles triangle is 7, find the radius of the circle drawn outside it.** - 2√7 cm - 3√7 cm - √6/6 cm - 2√6/6 cm - 5√7 cm **39) If the area of an isosceles triangle is , find the radius of the inscribed circle.** - √3/3 cm - √3/2 cm - 8√7 cm - 3√3/2 cm - 7√2 cm **40) If the area of an isosceles triangle is 5, find the radius of the circle drawn outside it.** - 3√30/3 cm - 3√19/19 cm - 3√5/5 cm - 7√3/12 cm - 3√3 cm **41) If the area of an isosceles triangle is 6, find the radius of the inscribed circle.** - 2√3 cm - 3√2 cm - 7√2 cm - 9√2/2 cm - 7√3/2 cm **42) If the area of an isosceles triangle is 4, find its altitude.** - 6 cm - 3√7/7 cm - 5√3/3 cm - 6√7/7 cm - 2√3/3 cm **43) If the area of an isosceles triangle is find its altitude.** - √75/15 cm - √13/13 cm - √14/14 cm - √17/17 cm - √23/23 cm **44) The area of an isosceles triangle is 12. Find the radius of the circle drawn outside it.** - 4 cm - 5√3 cm - 5√2 cm - 5√3/2 cm - 2√5 cm **45) If the area of an isosceles triangle is 15, find the radius of the circle drawn outside it.** - 2√5 cm - 3√10/4 cm - 6√3 cm - 2√3 cm - 3√5/2 cm **46) If the area of an isosceles triangle is 5, find its side.** - 2√5 cm - 5√3/5 cm - 5√2 cm - 5√2 cm **47) If the area of an isosceles triangle is 7, find its side.** - 2√7 cm - 5√7 cm - 7√2 cm - 7√7 cm - 5√3 cm **48) If the area of an isosceles triangle is 8/3 (cm)², find its side.** - 4√2 cm - √3/2 cm - 4√3 cm - (2√2)/3 cm - 5/3 cm **49) If the area of an isosceles triangle is 10, find its altitude.** - 15√3/3 cm - 15√3 cm - 3√15/15 cm - 14√3/14 cm - 3√3 cm **50) If the radius of a circle inscribed in a right triangle is 2 cm, find the area of this triangle.** - 36√3 cm² - 4√4 cm² - 6√4/25 cm² - 3√16 cm² - 16√3 cm² **51) If the radius of the inscribed circle of a right triangle is 3 cm, find the area of this triangle.** - 54√3 cm² - 9√3/16 cm² - 18√3 cm² - 2√3/9 cm² - 9√3 cm² **52) If the radius of the inscribed circle of a right triangle is 3 cm, find the area of this triangle.** - 135√3 cm² - 4√3/11 cm² - 9√3/4 cm² - 4√3/9 cm² - 18√3 cm² **53) If an isosceles triangle has an area, find its side.** - 4√2 cm - √3 cm - 4√3 cm - 2√2 cm - 5√3 cm **54) If the radius of the circumscribed circle of a right triangle is 6 cm, find the area of the triangle.** - 81√3 cm² - 9√3/4 cm² - 60√3 cm² - 4√3/9 cm² - 9√3 cm² **55) If the radius of the inscribed circle of a right triangle is 5 cm, find the area of the triangle.** - 75√3 cm² - 5√3/4 cm² - 4√3/25 cm² - 50√3 cm² - 5√3/9 cm² **56) If the radius of the inscribed circle of a right triangle is 1 cm, find the area of the triangle.** - 3√3 cm - 2√3 cm - 7√3 cm - 9√3 cm - 6√3 cm