Difference between revisions of "2022 AMC 10B Problems/Problem 2"
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Knowing that <math>AP=3</math> ; <math>AB=5</math>, then we would know that <math>BP=4</math> because <math>\triangle APB</math> is a 3-4-5 right triangle (as stated in solution 1). | Knowing that <math>AP=3</math> ; <math>AB=5</math>, then we would know that <math>BP=4</math> because <math>\triangle APB</math> is a 3-4-5 right triangle (as stated in solution 1). | ||
− | We get the area of <math>\triangle | + | We get the area of <math>\triangle ABD</math> as 10 because of the area of the triangle > <math>bh/2</math>. |
Since we know that <math>\triangle ABD</math> ≅ <math>\triangle BDC</math> and that <math>ABCD=\triangle ABD + \triangle BDC</math>. That means we double the area of <math>\triangle ADB</math>, <math>10 \cdot 2 = \boxed{\textbf{(D) }20}</math>. | Since we know that <math>\triangle ABD</math> ≅ <math>\triangle BDC</math> and that <math>ABCD=\triangle ABD + \triangle BDC</math>. That means we double the area of <math>\triangle ADB</math>, <math>10 \cdot 2 = \boxed{\textbf{(D) }20}</math>. |
Revision as of 02:35, 18 November 2022
- The following problem is from both the 2022 AMC 10B #2 and 2022 AMC 12B #2, so both problems redirect to this page.
Problem
In rhombus , point
lies on segment
so that
,
, and
. What is the area of
? (Note: The figure is not drawn to scale.)
Solution
(Figure redrawn to scale.)
.
is a rhombus, so
.
is a 3-4-5 right triangle, so
.
The area of the rhombus .
~richiedelgado
Solution 2 (The Area Of A Triangle)
The diagram is the same as solution 1, just constrcuted a line at
When it comes to the sides of a rhombus, their opposite sides are congruent and parallel. Which means ≅
by the Alternate Interior Angles Theorem.
By knowing these statements, we get ≅
.
Knowing that ;
, then we would know that
because
is a 3-4-5 right triangle (as stated in solution 1).
We get the area of as 10 because of the area of the triangle >
.
Since we know that ≅
and that
. That means we double the area of
,
.
~ghfhgvghj10
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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 |
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 1 |
Followed by Problem 3 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.