In right triangle , is a right angle. The altitude from to hypotenuse meets at point , dividing the hypotenuse into and . What is the perimeter, in centimeters, of ?

Solution available with step-by-step explanation

SAT Right Triangles & Trigonometry Question 42 | Free SAT Prep | Aniko

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Question 42·Hard·Right Triangles and Trigonometry

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In right triangle $\triangle ABC$ , $\angle C$ is a right angle. The altitude from $C$ to hypotenuse $\overline{AB}$ meets $\overline{AB}$ at point $D$ , dividing the hypotenuse into $AD = 9\text{ cm}$ and $DB = 16\text{ cm}$ . What is the perimeter, in centimeters, of $\triangle ABC$ ?

Your answer

(Express the answer as an integer)

For right-triangle problems with an altitude dropped to the hypotenuse, immediately note that the hypotenuse is split into two segments and that the two smaller triangles are similar to the original triangle. Use the standard relationships $AC^2 = AD \cdot AB$ and $BC^2 = DB \cdot AB$ to get the legs quickly without drawing a detailed diagram. Once the legs and hypotenuse are known, add them for the perimeter. This approach avoids messy Pythagorean calculations and keeps the work efficient and organized.

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Start with the hypotenuse

You know the hypotenuse is split into two segments of 9 cm and 16 cm. What is the total length of the hypotenuse $AB$ ?

Use the special property of the altitude to the hypotenuse

Dropping an altitude from the right angle to the hypotenuse creates two smaller right triangles that are similar to the original triangle. Think about how side ratios in similar triangles can give you equations involving $AC$ , $BC$ , $AD$ , $DB$ , and $AB$ .

Form equations for the legs

From similarity, you can get relationships of the form "(leg) $^2$ = (hypotenuse segment) $\times$ (whole hypotenuse)". Use $AD$ with $AB$ to find $AC$ , and $DB$ with $AB$ to find $BC$ .

Finish with the perimeter

Once you know $AC$ and $BC$ along with $AB$ , add the three side lengths to find the perimeter.

Desmos Guide

Confirm the hypotenuse length

In Desmos, type 9+16 to confirm that the hypotenuse $AB$ is 25.

Compute the legs using the similarity formulas

To find $AC$ , type sqrt(9*25). To find $BC$ , type sqrt(16*25). These use the relationships $AC^2 = AD*AB$ and $BC^2 = DB*AB$ .

Compute the perimeter

In Desmos, type sqrt(9*25) + sqrt(16*25) + 25. The resulting value is the perimeter of $\triangle ABC$ in centimeters.

Step-by-step Explanation

Find the hypotenuse

The altitude from $C$ hits the hypotenuse $AB$ at $D$ , splitting $AB$ into $AD=9$ and $DB=16$ .

So the full hypotenuse is:

AB = AD + DB = 9 + 16 = 25

Use similar triangles to relate the sides

When you drop an altitude from the right angle to the hypotenuse in a right triangle, the two smaller triangles are similar to the original triangle.

That means:

$\triangle ABC \sim \triangle ACD$
$\triangle ABC \sim \triangle CBD$

From $\triangle ABC \sim \triangle ACD$ , match the corresponding sides:

$AC$ (leg of big triangle) corresponds to $AD$ (part of hypotenuse)
$AB$ (hypotenuse of big triangle) corresponds to $AC$ (hypotenuse of smaller triangle)

So the ratio is

\frac{AC}{AB} = \frac{AD}{AC}

Cross-multiply:

AC^2 = AD \cdot AB

Similarly, from $\triangle ABC \sim \triangle CBD$ we get

BC^2 = DB \cdot AB

Compute the lengths of the legs

Now plug in the known segment lengths into the formulas from similarity.

For side $AC$ :

AC^2 = AD \cdot AB = 9 \cdot 25 = 225

AC = \sqrt{225} = 15

For side $BC$ :

BC^2 = DB \cdot AB = 16 \cdot 25 = 400

BC = \sqrt{400} = 20

Add the three sides for the perimeter

The sides of $\triangle ABC$ are:

$AC = 15$
$BC = 20$
$AB = 25$

The perimeter is the sum of all three sides:

15 + 20 + 25 = 60

So, the perimeter of $\triangle ABC$ is $60$ centimeters.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation