Social networks

OpenProf

Unlock access to content

Select one option:

Free of charge

Unlock the chapter "Pythagoras’ theorem"

Unlock access to the solution of problems in this chapter.
Free of charge Price: 3 £

Continue »

Subscribe to OpenProf If you do not get a better grade, you get your money back!

Unlock access to all problems solutions.
Available from 8.25 £ / month.

Continue »

mathematics UK Year 11 Mathematics (KS4)

Solved problems for the chapter:

Pythagoras’ theorem

View the theory »

Quadratic equation

1. General form of the quadratic equation

1.1. Simplifying quadratic equations

1.1.1. Example of simplifying a quadratic equation

1.1.2. Generalisation of simplifying quadratic equation

2. Solving quadratic equations

2.1. Case D> 0

2.2. Case D = 0

2.3. Case D < 0

2.4. Solving special forms of quadratic equations by factorisation

2.4.1. Case b = 0 (generalisation)

2.4.2. Case b = 0 (practical example)

2.4.3. Case c = 0 (generalisation)

2.4.4. Case c = 0 (example)

2.4.5. Case a = 1 (generalisation)

2.4.6. Case a = 1 (example)

3. Zero form of the quadratic equation

4. Forum

View the theory »

Quadratic function

1. Formats of the quadratic function

1.1. General form and meaning of the leading coefficient a

1.2. Zero shape and zero square functions

1.3. Top shape and quadratic function themes

1.3.1. Specify the peak shape

1.3.2. Determining the apex shape by completing it to a perfect square

2. Draw a graph of a quadratic function (parabola)

2.1. Drawing a parabola based on a general shape

2.2. Drawing a parabola based on the vertex shape

3. Mutual position of parabola and line

4. Forum

↓ Show theory [2] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:pythagoras%E2%80%99_theorem/5349/

Example 1.

The diagram below shows a right-angled triangle ABC. The area of triangle ABC is 10.39 , and side b is 3.46 cm long. The ratio of the lengths of the perpendicular projections of side a and b to the hypotenuse is 3.00. Calculate the height and the perimeter of the triangle.

Example 2.

The diagram below shows a right-angled triangle ABC. The hypotenuse c of the triangle is 7.42 cm and the height of the hypotenuse is is 3 cm. Calculate the sides a and b and the area of the triangle.

The diagram below shows a right-angled triangle ABC:

Given that the area of the triangle is 11.12 , and side a is 3 cm, calculate the perimeter of the triangle.

Calculate the length of the side of a rhombus with diagonals and .

Given a triangle with sides , and , determine whether the triangle is right-angled.

Carefully study the figure below and use it to answer the following questions:

The length of the hypotenuse is .
The length of side is .
The length of side is .
The area of square B is .

Calculate the perimeter and area of the given triangles with the data:

The perimeter of an isosceles triangle with the base length is . Calculate the area of the triangle.

The table below contains data of three right-angled triangles. Complete the table.

10.

The pentagon below has sides a, b, c, d and e. The point b lies on the line BC and the point a lies on the line AB. Calculate the perimeter and area of the pentagon given the length of side:

11.

Calculate the length of the diagonals and sides of the kite in the sketch below:

12.

Calculate the lengths of all plane diagonals and the length of the diagonal of a square with sides a = 3 cm, b = 6 cm and c = 5 cm. Round the result to one decimal place.

13.

In a right angle triangle, the hypotenuse measures , and the side a is shorter than the side b. What is the measurement of a and b.

14.

In a right angle triangle, the sum of the lengths of the legs is and the hypotenuse measures . Calculate the length of the legs.

15.

The sides of a right angle triangle measures . If is the hypotenuse, find the length of all sides.

16.

Given a square with sides . Determine the sides of the the inscribed rectangle that has sides parallel to the diagonals of the square and has the largest area.

17.

At the wall, as shown in the picture, a garden is enclosed with a 32 m long wire fence in the shape of an isosceles triangle, so that the area of the garden is the largest. What are the dimensions of the garden?

18.

Calculate the length of the unknown side x in the triangle.

19.

The sum of the base and height of a right angle triangle is 54 cm. Determine the length of these sides so that the hypotenuse of the triangle is minimum.

20.

The perimeter of an equilateral triangle is . Using the Pythagorean theorem, find the height of the triangle.

21.

For each example of a right-angled triangle in the figure below, write all three forms of Pythagoras' theorem.

22.

Calculate the width of a rectangle long with diagonal .

23.

Calculate the length of the hypotenuse and the perimeter in a right-angled triangle given the base as and the height as .

24.

The area of the square measures . Calculate the length of the diagonal of the square.

25.

We have an isosceles trapezoid with the bases and and the sides . Calculate the height of the given trapezoid .

26.

The figure below is a square with a given diagonal. Using the Pythagorean theorem, calculate the length of the square.

27.

Calculate missing sides x and y in the figure below:

28.

The sum of the lengths of the sides of a right angle triangle is , the length of its hypotenuse is . Calculate the lengths of the sides.

29.

The sum of the two sides of a right-angle triangle is , the length of its hypotenuse is . Calculate the length of the sides.