In , is a right angle. Point lies on such that is an altitude to . If and , which choice is the length of ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 51 | Free SAT Prep | Aniko

00:00

Question 51·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

0 / 200

In $\triangle ABC$ , $\angle B$ is a right angle. Point $D$ lies on $\overline{AC}$ such that $\overline{BD}$ is an altitude to $\overline{AC}$ . If $BD=120$ and $\sin A=\frac{12}{13}$ , which choice is the length of $\overline{DC}$ ? (Aniкο‌.аi‍)

When a trig ratio is a clean fraction like $\frac{12}{13}$ , treat it as a scaled Pythagorean triple: here $\sin A=\frac{BC}{AC}$ implies $BC:AB:AC=12:5:13$ . Then use a right-triangle altitude fact to connect the given altitude to the scale factor (either $BD=\frac{AB\cdot BC}{AC}$ or similarity), and finally use a projection formula such as $DC=\frac{BC^2}{AC}$ to get the requested hypotenuse segment. ©⁠ an⁠іko.⁠аi

Hints

Click me!

Turn $\sin A$ into a side ratio

Because $\angle B$ is $90^\circ$ , identify the hypotenuse and write $\sin A$ as a ratio of two side lengths.

Use the implied Pythagorean triple

If $\frac{BC}{AC}=\frac{12}{13}$ , then the third side is proportional to $5$ (since $5^2+12^2=13^2$ ). © Anіko

Connect the altitude to the side lengths

For a right triangle, the altitude from the right angle to the hypotenuse can be written in terms of the legs and the hypotenuse, or you can use similar triangles formed by the altitude.

Desmos Guide

Model the side lengths with a scale factor

Let $x$ represent the scale factor $k$ . Enter expressions for the sides: AB=5x, BC=12x, and AC=13x (you can just type 5x, 12x, and 13x as needed).

Solve for the scale factor using the altitude relationship

Graph y=(5x*12x)/(13x) and y=120. Click their intersection and note the $x$ -value (this is $k$ ). Wrіtt‍en ⁠by Аnik⁠o

Compute $DC$ from $k$

Enter DC=(12x)^2/(13x) and evaluate it at the $x$ -value you found (you can use a table or just read the value when $x$ equals that intersection). The resulting value is $DC$ .

Step-by-step Explanation

Use the sine ratio to set side lengths

Since $\angle B$ is a right angle, $AC$ is the hypotenuse. Also,

$\sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AC}=\frac{12}{13}$ .

So let $BC=12k$ and $AC=13k$ . Then

AB=\sqrt{AC^2-BC^2}=\sqrt{(13k)^2-(12k)^2}=\sqrt{169k^2-144k^2}=5k.

Use the altitude formula to find the scale factor

In a right triangle, the altitude from the right angle to the hypotenuse has length Тhiѕ ⁠questiοn ‌іs‍ frоm Aniко

BD=\frac{AB\cdot BC}{AC}.

Substitute $AB=5k$ , $BC=12k$ , and $AC=13k$ :

120=\frac{(5k)(12k)}{13k}=\frac{60k}{13}.

So $60k=1560$ , giving $k=26$ . (You can also get this from similar triangles if you don’t remember the formula.)

Find $DC$ using the projection relationship

In a right triangle with altitude to the hypotenuse, the segment adjacent to a leg satisfies

DC=\frac{BC^2}{AC}.

Compute with $BC=12k$ and $AC=13k$ :

DC=\frac{(12k)^2}{13k}=\frac{144k}{13}.

With $k=26$ ,

DC=\frac{144\cdot 26}{13}=144\cdot 2=288.

So the correct choice is 288.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation