HMMO

Geometry Difficulty 3.7 AMC 10/12 Find the answer United States

Problem:

In the Cartesian plane, a line segment with midpoint $(2020,11)$ has one endpoint at $(a, 0)$ and the other endpoint on the line $y = x$ . Compute $a$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Let the other endpoint be $(t, t)$ . The midpoint of $(a, 0)$ and $(t, t)$ is $\left(\frac{a + t}{2}, \frac{t}{2}\right)$ . So, we know that $\frac{a + t}{2} = 2020$ and $\frac{t}{2} = 11$ .

The second equation yields $t = 22$ . Substituting this into the first yields $a = 2 \cdot 2020 - 22 = 4018$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.