9.3 Use properties of angles, triangles, and the pythagorean theorem  (Page 4/15)

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We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

$\text{Δ}ABC$ and $\text{Δ}XYZ$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle. Solution

 Step 1. Read the problem. Draw the figure and label it with the given information. The figure is provided. Step 2. Identify what you are looking for. The length of the sides of similar triangles Step 3. Name. Choose a variable to represent it. Let a = length of the third side of $\Delta ABC$ y = length of the third side $\Delta XYZ$ Step 4. Translate. The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$ Since the side $AB=4$ corresponds to the side $XY=3$ , we will use the ratio $\frac{\mathrm{AB}}{\mathrm{XY}}=\frac{4}{3}$ to find the other sides. Be careful to match up corresponding sides correctly. Step 5. Solve the equation. Step 6. Check: Step 7. Answer the question. The third side of $\Delta ABC$ is 6 and the third side of $\Delta XYZ$ is 2.4.

$\text{Δ}ABC$ is similar to $\text{Δ}XYZ.$ Find $a.$ 8

$\text{Δ}ABC$ is similar to $\text{Δ}XYZ.$ Find $y.$ 22.5

Use the pythagorean theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around $500$ BCE.

Remember that a right triangle has a $\text{90°}$ angle, which we usually mark with a small square in the corner. The side of the triangle opposite the $\text{90°}$ angle is called the hypotenuse    , and the other two sides are called the legs . See [link] . In a right triangle, the side opposite the 90° angle is called the hypotenuse and each of the other sides is called a leg.

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

The pythagorean theorem

In any right triangle $\text{Δ}ABC,$

${a}^{2}+{b}^{2}={c}^{2}$

where $c$ is the length of the hypotenuse $a$ and $b$ are the lengths of the legs. To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation $\sqrt{m}$ and defined it in this way:

$\text{If}\phantom{\rule{0.2em}{0ex}}m={n}^{2},\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\sqrt{m}=n\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}n\ge 0$

For example, we found that $\sqrt{25}$ is $5$ because ${5}^{2}=25.$

We will use this definition of square roots to solve for the length of a side in a right triangle.

Use the Pythagorean Theorem to find the length of the hypotenuse. Solution

 Step 1. Read the problem. Step 2. Identify what you are looking for. the length of the hypotenuse of the triangle Step 3. Name. Choose a variable to represent it. Let $c=\text{the length of the hypotenuse}$ Step 4. Translate. Write the appropriate formula. Substitute. Step 5. Solve the equation. Step 6. Check: Step 7. Answer the question. The length of the hypotenuse is 5.

Use the Pythagorean Theorem to find the length of the hypotenuse. 10

Use the Pythagorean Theorem to find the length of the hypotenuse. 17

Use the Pythagorean Theorem to find the length of the longer leg. Solution

 Step 1. Read the problem. Step 2. Identify what you are looking for. The length of the leg of the triangle Step 3. Name. Choose a variable to represent it. Let $b=\text{the leg of the triangle}$ Label side b Step 4. Translate. Write the appropriate formula. Substitute. Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify. Step 6. Check: Step 7. Answer the question. The length of the leg is 12.

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