Question

3. A right triangle is shown.
D
18
7
G
Select all of the expressions that represent the
length of OG.
a)
√373
b) 25√11
c) 5√11
d) √18² + 72
e) 3
f) √18-7
g) √275
h) √182 - 72
i) Approx. 19.3
j) Approx. 16.6
Pleaseee help

Answer 1

Answer:

$\textsf{c)}\quad 5\sqrt{11}$

$\textsf{g)}\quad \sqrt{275}$

$\textsf{h)}\quad \sqrt{18^2-7^2}$

$\textsf{j)}\quad \sf Approx.\; 16.6$

Step-by-step explanation:

To find the length of side OG in right triangle ODG, we can use the Pythagorean Theorem:

$\boxed{\begin{array}{l}\underline{\textsf{Pythagorean Theorem}}\\\\a^2+b^2=c^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{ a and b are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{ c is the hypotenuse (longest side) of the right triangle.}\\\end{array}}$

In this case:

• a = OD = 7
• b = OG
• c = DG = 18

Substitute the values into the formula:

$7^2+OG^2=18^2$

Solve for OG:

$OG^2=18^2-7^2\\\\\\OG=\sqrt{18^2-7^2}\\\\\\OG=\sqrt{324-49}\\\\\\OG=\sqrt{275}\\\\\\OG=16.5831239517...\\\\\\OG\approx 16.6\; \sf (nearest\;tenth)$

Therefore, the exact length of OG is $\sqrt{275}$, which is approximately 16.6 (rounded to the nearest tenth).

We can simplify the radical by rewriting 275 as the product of its prime factors:

$275 = 5 \times 5 \times 11\\\\275=5^2 \times 11$

Therefore:

$\sqrt{275}=\sqrt{5^2 \times 11}$

$\textsf{Apply the radical rule:} \quad \sqrt{ab}=\sqrt{\vphantom{b}a}\sqrt{b}$

$\sqrt{275}=\sqrt{5^2}\sqrt{11}\\\\\sqrt{275}=5\sqrt{11}$

So, the expressions that represent the length of OG are:

$\textsf{c)}\quad 5\sqrt{11}$

$\textsf{g)}\quad \sqrt{275}$

$\textsf{h)}\quad \sqrt{18^2-7^2}$

$\textsf{j)}\quad \sf Approx.\; 16.6$