2024 AMC 12A Problems/Problem 5

The following problem is from both the 2024 AMC 12A #5 and 2024 AMC 10A #7, so both problems redirect to this page.

Problem

Let $M$ be the midpoint of segment $\overline{AB}$ , and let $T$ lie on segment $\overline{AB}$ so that $AT \cdot AM = 100$ and $BT \cdot BM = 28$ . What is the length of segment $\overline{TM}$ ?

$\textbf{(A)}~4\qquad \textbf{(B)}~4.5\qquad \textbf{(C)}~5 \qquad \textbf{(D)}~5.5 \qquad \textbf{(E)}~6$

Solution

Note that $AM = BM$ , so $\tfrac{AT}{BT} = \tfrac{100}{28} = \tfrac{25}{7}$ . Thus, let $AT = 25x$ and $BT = 7x$ . Then $AM = BM = 16x$ and $TM = 9x$ . We can now use either of the two conditions presented, WLOG we use the first: $AT \cdot AM = (25x)(16x) = 400x^{2} = 100 \implies x = 0.5$ Therefore, $TM = 9x = \boxed{\textbf{(B)}~4.5}$ .

2024 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2024 AMC 12A Problems/Problem 5

Problem

Solution

See also