Question

The sides of a triangle are $30,$ $70,$ and $80$ units. If an altitude is dropped upon the side of length $80,$ the larger segment cut off on this side is:

$62$

$63$

$64$

$65$

$66$

Answer: D

Concepts:altitude Pythagorean Theorem right triangle

Difficulty rating: 1550

Small Hint:

Let the altitude divide the side of length $80$ into $x$ and $80-x$

Big Hint:

Equate the two expressions for the square of the altitude

Solution:

Let $x$ be the segment adjacent to the side of length $30.$ If the altitude has length $h,$ then $\begin{aligned} 30^2-x^2&=h^2,\\ h^2&=70^2-(80-x)^2. \end{aligned}$ Cancelling $x^2$ and solving gives $x=15.$ The other segment is $80-15=65,$ which is the larger one.

Thus, the correct answer is D.

1958 AMC 12 Problem 36

Answer: D

Difficulty rating: 1550

Small Hint:

Big Hint:

Solution:

Problem 36 in Other Years