Pythagoras' Theorem and Triangle Calculations
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Questions and Answers

What is the length of RT in triangle RST if RS is 50cm and ST is 2m?

  • 1.9m
  • 1.8m (correct)
  • 1.5m
  • 2.6m

How can Belinda show that her triangle is right-angled?

  • Count the number of sides.
  • Check if all sides are equal.
  • Use the Pythagorean theorem to check the sides. (correct)
  • Measure the angles of the triangle.

Given a triangle with sides measuring 4cm, 6cm, and 7cm, is it a right-angled triangle?

  • No, because the longest side is not the hypotenuse.
  • Yes, because one side is twice the shortest side.
  • Yes, because 4² + 6² = 7².
  • No, as it does not satisfy the Pythagorean theorem. (correct)

What is the height of the tree if the distance from a point on the ground to the base is 18 metres and to the top is 20 metres?

<p>16.0m (A)</p> Signup and view all the answers

What is the length of the diagonal x of a rectangle if the length and width are both unknown?

<p>It can be calculated if the length and width are given. (C)</p> Signup and view all the answers

What is the area of rectangle ABCD if AD measures 5cm and BD measures 13cm?

<p>65cm² (D)</p> Signup and view all the answers

What is the height, h, of an isosceles triangle ABC where AB = BC = 8cm and AC = 5cm?

<p>4.0cm (D)</p> Signup and view all the answers

How far does the helicopter fly in total if it flies 60 miles west and then 25 miles north?

<p>85.0 miles (C)</p> Signup and view all the answers

What is the formula represented by Pythagoras’ theorem?

<p>$a^2 + b^2 = c^2$ (D)</p> Signup and view all the answers

In a right-angled triangle where AC = 8 cm and BC = 15 cm, what is the length of AB?

<p>13 cm (C)</p> Signup and view all the answers

If AC = 6 cm and AB = 20 cm in a right-angled triangle, what is the length of BC to one decimal place?

<p>18.8 cm (C)</p> Signup and view all the answers

To find the perimeter of a right-angled triangle with sides 7 cm, 24 cm, and the hypotenuse, which of the following is correct?

<p>31 cm (C)</p> Signup and view all the answers

What is the area of a right-angled triangle with base AC = 8 cm and height BC = 15 cm?

<p>90 cm² (C)</p> Signup and view all the answers

Given a ladder of 4 meters placed 1.5 meters away from a wall, how far does it reach up the wall?

<p>3.5 m (C)</p> Signup and view all the answers

What mistake could Charles have made to conclude that the missing side is 41 cm?

<p>Mistaking the hypotenuse for one of the legs (C)</p> Signup and view all the answers

In a right-angled triangle with sides 7 cm, 24 cm, what is the length of the hypotenuse AB?

<p>25 cm (A)</p> Signup and view all the answers

What is the relationship between the length and width of the rectangular field for the dog park?

<p>The length is twice the width. (C)</p> Signup and view all the answers

How do you calculate the area of the equilateral triangle with a side length of 4 cm?

<p>$\frac{\sqrt{3}}{4} \cdot 4^2$ (C)</p> Signup and view all the answers

What is the total length of wire required to make the trapezium frame if all side lengths are equal to x cm?

<p>4x (B)</p> Signup and view all the answers

If the mass of the wire is 0.489 g per cm, what would be the mass of a metal frame made from 5 pieces of wire, each 10 cm long?

<p>24.45 g (B)</p> Signup and view all the answers

In the right-angled triangle, if the perimeter is equal to the circumference of the circle, what does this imply about their dimensions?

<p>The sum of the side lengths equals the circle's circumference. (D)</p> Signup and view all the answers

What is the likely method to find the length of line LM joining the points L(−3, 2) and M(1, −3)?

<p>Applying the distance formula. (D)</p> Signup and view all the answers

To calculate the length of the wall dividing the two sections in the dog park, what additional information is essential?

<p>The width of the field. (B)</p> Signup and view all the answers

What would you expect the value of x to represent in the context of the right-angled triangle and circle problem?

<p>The diameter of the circle. (A)</p> Signup and view all the answers

Flashcards

Hypotenuse

The longest side of a right-angled triangle, opposite the right angle.

Pythagorean Theorem

A formula used to find the relationship between the sides of a right-angled triangle: a² + b² = c² where 'c' is the hypotenuse and 'a' and 'b' are the other two sides.

Isosceles triangle

A triangle with two sides of equal length and two equal angles.

Height of an isosceles triangle

The perpendicular distance from the base of an isosceles triangle to the opposite vertex (the point where the two equal sides meet).

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Diagonal of a rectangle

A line segment that connects two opposite vertices of a rectangle.

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Area of a rectangle

The area of a rectangle is calculated by multiplying its length and width: Area = length x width.

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Distance Formula

The length of a line segment is the distance between its two endpoints. To find the length of a line segment given its endpoints, you can use the distance formula, which is derived from the Pythagorean theorem.

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Calculate line length

The length of a line segment can be calculated using the distance formula, which takes into account the coordinates of its endpoints.

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Perimeter of a Trapezium

The perimeter of a shape is the total length of its sides. In the context of a trapezium, the perimeter is calculated by summing the lengths of all its sides.

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Area of an Equilateral Triangle

The area of a triangle is calculated by multiplying the base length by the height and dividing the result by 2. For an equilateral triangle, all sides are equal, and the height is equal to the square root of 3 divided by 2, multiplied by the side length.

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Circumference of a Circle

The circumference of a circle is the total length of the boundary of the circle. It can be calculated by using the formula C = 2πr, where C is the circumference, π is approximately 3.14159, and r is the radius of the circle.

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Area of a Triangle

The area of a triangle is half the product of its base and height. The height is perpendicular to the base, and it can be calculated using the Pythagorean theorem if the triangle is right-angled.

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Perimeter of a Triangle

The perimeter of a shape is the total length of its sides. In the context of a triangle, the perimeter is calculated by summing the lengths of all its sides.

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What is the hypotenuse?

The longest side of a right-angled triangle is called the hypotenuse.

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What is the Pythagorean Theorem?

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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How to find the missing side of a right-angled triangle?

To find the length of the missing side, use the Pythagorean Theorem: a² + b² = c². Substitute the known values, solve for the unknown side.

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How to find the perimeter of a triangle?

The perimeter of a triangle is the total length of all its sides. Add the lengths of all three sides to find the perimeter.

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How to find the area of a triangle?

To find the area of a triangle, multiply the base by the height and divide by 2. The height is the perpendicular distance from the base to the opposite vertex.

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What is a common mistake in applying the Pythagorean Theorem?

A mistake in calculations can be made by incorrectly applying the Pythagorean Theorem. Double check your calculation and make sure you're using the correct values.

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How can you find the area of a right-angled triangle given only the base and hypotenuse?

The area of a triangle is half the product of its base and height. Use the Pythagorean Theorem first to find the height, then calculate the area.

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How to find the length of the missing side of a right-angled triangle?

To find the length of the missing side, use the Pythagorean Theorem: a² + b² = c². Substitute the known values, solve for the unknown side.

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Study Notes

Pythagoras' Theorem

  • Used to find missing sides in right-angled triangles
  • Formula: a² + b² = c²
    • a and b represent the lengths of the two shorter sides (legs)
    • c represents the length of the longest side (hypotenuse)

Right-Angled Triangle Calculations

  • Various examples provided of applying Pythagoras' theorem to different right-angled triangles
  • Calculations involved finding missing sides (using the theorem)
  • Measurements given in centimetres (cm) or metres (m)

Perimeter Calculations

  • Perimeter is the total distance around the outside of a shape.
  • Calculations to find the perimeter of different shapes (including triangles) are in the examples
  • Answer should be provided to a specific decimal place (e.g., 1 decimal place)
  • Calculations in centimetres (cm) or metres (m).

Area Calculations

  • Area is calculated for right-angled triangles
  • Formula used for the area of a triangle: area = ½ * base * height
  • Calculations using figures in centimetres (cm) to calculate the area is given, and all figures are in cm

Isosceles Triangles

  • Triangles with two equal sides
  • Calculations to find the height of an isosceles triangle provided

Other Calculations

  • Calculations involving various aspects of different shapes, with detailed context (e.g., distance from a point to the top of a tree, to find height) in problems.
  • Calculations in metres (m), miles (miles), centimetres (cm).
  • Calculations on plane shapes provided, including rectangles, trapezoids, and equilateral triangles.

Coordinate Geometry

  • Calculations involving finding the length of a line segment connecting two points on a coordinate plane

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Pythagoras GCSE Maths Past Paper PDF

Description

This quiz covers the application of Pythagoras' theorem to find missing sides in right-angled triangles, as well as calculating perimeter and area for various shapes. Learn how to use the formulas effectively and practice with examples involving measurements in centimetres and metres.

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