Question

Triangle ABC has AB = BC = 1 and ∠ABC = 90°. Points P and Q are on two sides of the triangle so that PQ cuts triangle ABC into two pieces of equal area. If P is on AB and Q is on BC, with:

Answer 1

In triangle ABC, with AB = BC = 1 and ∠ABC = 90°, the coordinates of points P and Q, which cut triangle ABC into two equal area pieces, are (0, x) and (x, 0) respectively.

Step-by-step explanation:

Triangle ABC is a right triangle with AB = BC = 1 and ∠ABC = 90°. Let's assume that P is the point on AB and Q is the point on BC such that PQ cuts triangle ABC into two equal area pieces. We need to find the coordinates of P and Q.

Since AB = BC = 1, we can assume that BC is the x-axis and AB is the y-axis. Let's denote the coordinates of P as (0, y) and the coordinates of Q as (x, 0).

The area of triangle ABC can be found using the formula: Area = 0.5 * base * height.

Since triangle ABC is a right triangle, the base is AB = 1 and the height is BC = 1.

Therefore, the area of triangle ABC is 0.5 * 1 * 1 = 0.5.

The area of triangle ABP can be found using the same formula: Area = 0.5 * base * height.

The base is AP = 1 and the height is BP = y.

The area of triangle ABP is 0.5 * 1 * y = 0.5y.

The area of triangle BCQ can also be found using the same formula: Area = 0.5 * base * height.

The base is BQ = x and the height is CQ = 1.

The area of triangle BCQ is 0.5 * x * 1 = 0.5x.

Since the two triangles ABP and BCQ have equal areas, we can set up the following equation: 0.5y = 0.5x.

This equation simplifies to y = x.

Therefore, the coordinates of P are (0, x) and the coordinates of Q are (x, 0).