Question

A) Create an inequality involving sides AD and AC based on the information provided.

b) Create a sum involving side lengths from ∆ABC that is equal to the length of line segment AD. Explain your reasoning.
c) Combine the answers from parts a and b to relate the side lengths of ∆ABC. What theorem does this construction and logic prove?
A) Triangle Inequality Theorem
B) Law of Cosines
C) Pythagorean Theorem
D) Angle Bisector Theorem

Answer 1

The student's question requires the application of the Pythagorean theorem to relate the sides of a right triangle and solve for inequalities and sums involving these side lengths. The logic and constructions given in the answers affirm the Pythagorean theorem.

Step-by-step explanation:

The student's question involves solving problems using the Pythagorean theorem, which relates the lengths of the sides in a right triangle. The theorem can be stated as a² + b² = c², where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse.

Part a)

We need to create an inequality involving sides AD and AC in ∆ABC. If ∆ABC is a right triangle with AD and AC as the legs and DC as the hypotenuse, we can use the Pythagorean theorem to write an inequality AD² + AC² < DC², assuming D is a point on side BC, making ADC a right triangle.

Part b)

For a sum involving the side lengths of ∆ABC equal to the length of line segment AD, we could say AD = √(AC² + BC²), assuming ∆ABC is a right triangle with angle BAC as the right angle. This is due to the Pythagorean theorem asserting that in a right triangle, the leg is equal to the square root of the sum of the squares of the other two sides.

Part c)

Combining the answers from parts a) and b), we can relate the side lengths of ∆ABC by stating that AD² + AC² < DC² = √(AC² + BC²). This adheres to the Pythagorean theorem, which this construction and logic prove (Option C).