Question

cp geometry
similar right triangles day 1
assignment #2
name
period
date
find the missing length indicated. leave your answer in simplest radical form.
1)
2)
3)
4)
5)
6)
7)
8)

Explanation:

For all these problems, we use geometric properties of similar right triangles:

1. When an altitude is drawn to the hypotenuse of a right triangle:

• The length of the altitude $x$ is the geometric mean of the segments of the hypotenuse: $x = \sqrt{ab}$ (for hypotenuse split into $a$ and $b$)
• The length of a leg is the geometric mean of the hypotenuse and the adjacent hypotenuse segment: $leg = \sqrt{c \times adjacent\ segment}$ (where $c$ is full hypotenuse length)
• The altitude to hypotenuse: $x = \frac{leg_1 \times leg_2}{hypotenuse}$

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Problem 1

Step1: Use altitude-hypotenuse formula

$x = \frac{\sqrt{100 \times 36} \times \sqrt{100 \times (100-36)}}{100}$
Simplify: $x = \frac{\sqrt{100 \times 36} \times \sqrt{100 \times 64}}{100} = \frac{60 \times 80}{100}$

Step2: Calculate final value

$x = \frac{4800}{100} = 48$

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Problem 2

Step1: Find full hypotenuse length

Full hypotenuse $= 25$
Adjacent segment to $x$: $25 - 9 = 16$

Step2: Use leg-hypotenuse formula

$x = \sqrt{25 \times 16}$
$x = \sqrt{400} = 20$

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Problem 3

Step1: Use altitude-hypotenuse formula

$x = \sqrt{9 \times (25-9)}$
$x = \sqrt{9 \times 16}$

Step2: Calculate final value

$x = \sqrt{144} = 12$

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Problem 4

Step1: Find full hypotenuse length

Full hypotenuse $= 81$
Adjacent segment to $x$: $81 - 45 = 36$

Step2: Use leg-hypotenuse formula

$x = \sqrt{81 \times 36}$
$x = 9 \times 6 = 54$

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Problem 5

Step1: Calculate hypotenuse length

Hypotenuse $= 7 + 9 = 16$
Legs: $\sqrt{7^2 + x^2}$ and $\sqrt{9^2 + x^2}$
Use Pythagoras: $(\sqrt{7^2 + x^2})^2 + (\sqrt{9^2 + x^2})^2 = 16^2$
Simplify: $49 + x^2 + 81 + x^2 = 256$

Step2: Solve for $x$

$2x^2 = 256 - 130 = 126$
$x^2 = 63$
$x = \sqrt{63} = 3\sqrt{7}$

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Problem 6

Step1: Calculate hypotenuse length

Hypotenuse $= 84 + 16 = 100$
Legs: $\sqrt{84^2 + x^2}$ and $\sqrt{16^2 + x^2}$
Use Pythagoras: $(\sqrt{84^2 + x^2})^2 + (\sqrt{16^2 + x^2})^2 = 100^2$
Simplify: $7056 + x^2 + 256 + x^2 = 10000$

Step2: Solve for $x$

$2x^2 = 10000 - 7312 = 2688$
$x^2 = 1344$
$x = \sqrt{1344} = 8\sqrt{21}$

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Problem 7

Step1: Use altitude-hypotenuse formula

$12 = \sqrt{x \times 16}$
Square both sides: $144 = 16x$

Step2: Solve for $x$

$x = \frac{144}{16} = 9$

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Problem 8

Step1: Find other leg first

Other leg $= \sqrt{64^2 - 48^2} = \sqrt{(64-48)(64+48)} = \sqrt{16 \times 112} = \sqrt{1792} = 16\sqrt{7}$

Step2: Use altitude-hypotenuse formula

$48 = \frac{16\sqrt{7} \times x}{64}$
$48 = \frac{x\sqrt{7}}{4}$
$x = \frac{192}{\sqrt{7}} = \frac{192\sqrt{7}}{7}$

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Answer:

1. $\boldsymbol{48}$
2. $\boldsymbol{20}$
3. $\boldsymbol{12}$
4. $\boldsymbol{54}$
5. $\boldsymbol{3\sqrt{7}}$
6. $\boldsymbol{8\sqrt{21}}$
7. $\boldsymbol{9}$
8. $\boldsymbol{\frac{192\sqrt{7}}{7}}$