Let the bottom edge of a rectangular mirror on a vertical wall be parallel to and h feet above the level floor. If a person with eyes t feet above the floor is standing direct at a distance d feet from the mirror, what is the relationship among h,d and t if the person can just see his own feet in the mirror?

Question

Let the bottom edge of a rectangular mirror on a vertical wall be parallel to and h feet above the level floor. If a person with eyes t feet above the floor is standing direct at a distance d feet from the mirror, what is the relationship among h,d and t if the person can just see his own feet in the mirror?

Choose one answer.

a. t=4d and h does not matter.

b. h^2 + d^2 = (t^2)/4

c. t - h = d

d. (t - h) = 4d

e. t=2h and d does not matter.

Answer 1

(e)

The bottom of the mirror must be halfway up t, so the angles to and from the mirror are equal

Answer 2

To solve this problem, let's consider the situation. We have a rectangular mirror on a vertical wall, and the bottom edge of the mirror is h feet above the level floor. A person with eyes t feet above the floor is standing directly in front of the mirror at a distance d feet.

The person can just see their own feet in the mirror. This means that the line of sight from the person's eyes to their feet should just touch the top edge of the mirror.

Let's draw a diagram to visualize this.

+------------+
| |
| |
| person |
| |
| |
+------------+ mirror
| |
h | |
| |
| |
__________|__________|__________ floor

Now, let's analyze the situation. The person is standing at a distance of d feet from the mirror. The line connecting the person's eyes to their feet forms a right triangle with the mirror. The distance from the person's eyes to the mirror is t - h (the height of the person minus the height of the mirror).

If we consider this right triangle, we can use the Pythagorean theorem to find a relationship among h, d, and t.

According to the Pythagorean theorem: In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

So, we have:

(hypotenuse)^2 = (side1)^2 + (side2)^2

In our case:

(t - h)^2 = d^2 + h^2

Expanding and simplifying, we get:

t^2 - 2th + h^2 = d^2 + h^2

Rearranging the equation, we get:

t^2 - 2th = d^2

Dividing both sides by 4, we get:

(t^2 - 2th)/4 = d^2/4

Simplifying further, we have:

(t - h) = d^2/4

Therefore, the correct relationship among h, d, and t if the person can just see their own feet in the mirror is:

d. (t - h) = 4d