What theorem relates the sides of a right-angled triangle?

In a right-angled triangle, what is the relationship between the hypotenuse and the other two sides according to the Pythagoras Theorem?

Pythagoras Theorem

The square of the hypotenuse equals the sum of the squares of the other two sides.

Which mathematician is primarily credited with the Pythagorean Theorem?

Which mathematician is credited with giving the Pythagoras Theorem its modern form?

Pythagoras

Pythagoras

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. True or False?

True or False: The Pythagorean triplet 5, 12, 13 satisfies the equation a² + b² = c².

True

True

Fill in the blank: In triangle ABC with a right angle at B, ______ = AB² + BC².

Fill in the blank: In triangle ABC, if AB is the largest side and AB² = AC² + BC², then ∠C = ______.

AC²

90°

Which of the following statements correctly describes the hypotenuse in a right-angled triangle? A) It is the shortest side B) It is the longest side C) It is opposite the smallest angle D) It is always equal to the other two sides

What is the formula used to express the Pythagoras Theorem in triangle ABC?

B) It is the longest side

AC² = AB² + BC²

Riddle: I am a triangle with one angle that is exactly 90 degrees, what type of triangle am I?

Riddle: I relate three sides in a triangle, but I'm not a person. What am I?

Right-angled triangle

Pythagoras Theorem

If AB is the largest side in triangle ABC and AB² = AC² + BC², what can we conclude about angle C?

If AB² > AC² + BC², what type of triangle is formed at angle ACB?

Angle C is 90 degrees.

Obtuse-angled triangle

Fill in the blank: A set of three positive integers a, b, c that satisfy the equation a² + b² = c² are called ______.

True or False: The hypotenuse is always the smallest side in a right-angled triangle.

Pythagorean triplets

False. The hypotenuse is always the largest side in a right-angled triangle.

True or False: If AB² > AC² + BC² in triangle ABC, then angle ACB is acute.

In the area-based proof of the Pythagoras Theorem, what shape is formed on each side of the triangle?

False. Angle ACB is obtuse.

Squares

Which proof method involves proving triangles are similar by AA similarity?

Fill in the blank: If the square of the largest side of a triangle equals the sum of the squares of the other two sides, the triangle is ______.

Similarity-based proof

Right-angled

Complete the statement: In an isosceles triangle ABC with AB = AC, if E is the midpoint of BC and AD is perpendicular to BC, then ______ = 2BC × ED.

What is the formula derived from the Extended Pythagoras Theorem in triangle ABC with AD perpendicular to BC produced?

AB² - AC²

C² = a² + b² + 2ax

Multiple Choice: In a parallelogram, which of the following is true? A) AC² + BD² = AB² + BC² B) AC² + BD² > AB² + BC² C) AC² + BD² < ab²="" +="" bc²="" d)="" ac²="" +="" bd²="" is="" independent="" of="">

Riddle: I can be congruent or similar, but I'm not a shape. What concept am I related to in geometry?

A) AC² + BD² = AB² + BC²

Triangles

Fill in the blank: The relationship expressed in the extended Pythagorean theorem is c² = a² + b² + ______.

In the similarity-based proof of the Pythagoras Theorem, what does AA stand for?

2ax

Angle-Angle

Riddle: I can be found in triangles, my square connects the other two sides, who am I?

True or False: The area of a rectangle is equal to the sum of the areas of two right triangles formed by its diagonal.

Hypotenuse

True

What is the relationship between the areas of triangles and rectangles in the proof involving triangles GAC and BAE?

What is the relationship between the areas of triangles GAC and BAE in the area-based proof?

They have equal areas.

They have equal areas.

True or False: The area of square ABFG is equal to the area of rectangle AMNE.

Fill in the blank: In triangle ABC with ∠ACB = 90°, if CD is perpendicular to AB, then 1/p² = 1/a² + 1/b², where p is the length of CD. This is known as ______.

True

Pythagorean theorem for perpendiculars

What is the formula for the area of a triangle?

What is the condition for a triangle to be classified as acute-angled based on the sides?

Area = ½ × base × height

If AB² < ac²="" +="">

Fill in the blank: In triangle ABC, if angle ACB = 90°, then AB² = ______ + AC².

Riddle: I can be extended, but my basic form relates to squares and triangles. What am I?

BC²

Pythagoras Theorem

Which condition indicates that a triangle is right-angled based on the Pythagorean Theorem?

In a parallelogram, what is the relationship expressed between the diagonals and the sides?

The square of the largest side equals the sum of the squares of the other two sides.

AC² + BD² = AB² + BC² + CD² + DA²

Riddle: I am the largest side in a right triangle, and my square is the sum of the squares of the other two sides. What am I?

Which of the following statements accurately describes the Pythagorean theorem? A) The square of the hypotenuse equals the product of the other two sides. B) The square of the hypotenuse equals the sum of the squares of the other two sides. C) The sum of the angles in a triangle is 180°. D) The area of a triangle is half the product of its base and height.

Hypotenuse

B) The square of the hypotenuse equals the sum of the squares of the other two sides.

True or False: If the square of the largest side of a triangle equals the sum of the squares of the other two sides, the triangle is obtuse-angled.

False. If the square of the largest side equals the sum of the squares of the other two sides, the triangle is right-angled.

Fill in the blank: The formula for the Pythagorean theorem states that in a right-angled triangle, ______.

AC² = AB² + BC²

Riddle: I'm a relationship in geometry, connecting three sides, when the right angle resides. What am I?

Pythagorean theorem

What type of triangles can be classified based on the Pythagorean theorem? Provide all possible classifications.

Right-angled, obtuse-angled, acute-angled triangles.