Question

PROOF: Write a coordinate proof for the statement.

The measure of the segment that joins the vertex of the right angle in a right triangle to the midpoint of the hypotenuse is one-half the measure of the hypotenuse.

Given: Right (Triangle)ABC; P is the midpoint of BC
Prove: AP = 1/2 BC

(Find AP before finding BC)

Answer 1

By placing the right triangle on a coordinate plane and using the distance formula, we can show that the length of segment AP is equal to half the hypotenuse BC based on the Pythagorean theorem.

Step-by-step explanation:

To prove that the measure of the segment that joins the vertex of the right angle in a right triangle to the midpoint of the hypotenuse is one-half the measure of the hypotenuse, we can use the properties of right triangles and the Pythagorean theorem.

Let's place right triangle ABC on the coordinate plane with vertex A at the origin (0,0), vertex B on the x-axis, and vertex C on the y-axis, such that AB = x and AC = y. Since point P is the midpoint of hypotenuse BC, its coordinates will be ((x/2), (y/2)). Now, we want to find the length of segment AP.

Using the distance formula:
AP = √((x/2)^2 + (y/2)^2) = √(x^2/4 + y^2/4) = √(x^2 + y^2)/2
Because ABC is a right triangle, we apply the Pythagorean theorem:
x^2 + y^2 = BC^2

Therefore, AP = √(BC^2)/2 = BC/2. This shows that AP is indeed one-half the measure of the hypotenuse BC.