Rectangle Problem

Geometry Level 1

In the figure, $ABCD$ is a rectangle with side lengths $21$ and $28$ . Point $E$ is on $AC$ such that $AE = 30$ . What is the length of $DE$ ?

The answer is of the form $a\sqrt {b}$ . Input $a + b$ as your answer.

The answer is 68.

2 solutions

Chew-Seong Cheong
Aug 11, 2020

We note that $\triangle ABC$ is a $3$ - $4$ - $5$ right triangle. Therefore $AC=35$ and $CE = AC - AE = 5$ .

Draw $EF || BC$ and $EG || AB$ . Then we note that $\triangle CEF$ , $\triangle AEG$ and $\triangle ABC$ are similar. Then we have $\dfrac {EF}{CE} = \dfrac {BC}{AC} \implies EF = \dfrac {BC}{AC} \times CE = \dfrac {21}{35} \times 5 = 3$ . Similarly, $EG = \dfrac {28}{35} \times 30 = 24$ .

By Pythagorean theorem , we have $DE = \sqrt{EF^2+EG^2} = \sqrt{3^2+24^2} = 3\sqrt{65}$ .

And we have $a + b = 3 + 65 = \boxed{68}$ .

@Barry Leung , we don't need to use cm in Geometry problem. It doesn't add information to the problem. It should be "What is the length of $DE$ ? instead of "What is $DE$ ?". Because $DE$ is a line segment. Since you have used $A$ and $B$ as points it would be preferred to use small later $a$ and $b$ is the answer form. In fact small letters appeal in most of the questions in Brilliant.org.

Chew-Seong Cheong - 10 months ago

Marvin Kalngan
Aug 13, 2020

Rectangle Problem

The answer is 68.

2 solutions

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