Math / Geometry

Question16. Solve for $x$ and $y$ . Round to the tenths place if necessary.

Studdy Solution

STEP 1

1. The triangles are similar, which means their corresponding sides are proportional.
2. We need to solve for $x$ and $y$ .
3. The sides of the smaller triangle are 2, $x$ , and 6.
4. The sides of the larger triangle are $x + 3$ , $y$ , and 10.

STEP 2

1. Set up the proportion equations using the similarity of triangles.
2. Solve the proportion equations to find $x$ .
3. Use the value of $x$ to find $y$ .

STEP 3

Since the triangles are similar, the corresponding sides are proportional. We can set up the following proportion using the sides of the triangles:
$\frac{2}{x+3} = \frac{x}{y} = \frac{6}{10}$

STEP 4

First, solve for $x$ using the proportion between the sides of the smaller triangle and the hypotenuse:
$\frac{2}{x+3} = \frac{6}{10}$
Cross-multiply to solve for $x$ :
$2 \cdot 10 = 6 \cdot (x + 3)$

STEP 5

Simplify and solve the equation:
$20 = 6x + 18$
Subtract 18 from both sides:
$20 - 18 = 6x$
$2 = 6x$
Divide both sides by 6:
$x = \frac{2}{6} = \frac{1}{3} \approx 0.3$

STEP 6

Now, solve for $y$ using the proportion:
$\frac{x}{y} = \frac{6}{10}$
Substitute $x = 0.3$ :
$\frac{0.3}{y} = \frac{6}{10}$
Cross-multiply to solve for $y$ :
$0.3 \cdot 10 = 6 \cdot y$

STEP 7

Simplify and solve the equation:
$3 = 6y$
Divide both sides by 6:
$y = \frac{3}{6} = \frac{1}{2} = 0.5$
The values of $x$ and $y$ are:
$x \approx 0.3$ $y = 0.5$

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